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y(use)j(in)f(computations)h(are)h(normal)e(co)s(ordinates.)0 1106 y Fl(De\014nition)35 b(2.2.1)47 b Fp(The)41 b Fs(normal)31 b(co)s(ordinate)h(map)i Fp(of)h Fo(F)47 b Fp(is)35 b(the)f(map)i Fo(\027)2621 1120 y Fk(F)2708 1106 y Fs(:)28 b Fo(N)2834 1120 y Fk(F)2922 1106 y Fm(!)g Fl(R)3119 1073 y Fk(n)3200 1106 y Fp(de\014ne)-5 b(d)36 b(by)0 1218 y(the)d(formula)1391 1422 y Fo(\027)1436 1436 y Fk(F)1495 1422 y Fs(\()p Fo(x;)15 b(v)s Fs(\))26 b(=)f Fo(F)13 b Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s(:)0 1625 y Fs(This)29 b(is)g(a)i(di\013eomorphism)c(of)k (a)g(neigh)m(b)s(orho)s(o)s(d)d(of)i(the)h(zero)g(section)f(of)h Fo(N)2697 1639 y Fk(F)2756 1625 y Fs(.)136 1738 y(F)-8 b(or)40 b(future)e(reference)i(w)m(e)f(calculate)h(the)f(Jacobian)g (determinan)m(t)f(of)i(the)f(normal)f(co)s(ordinate)0 1851 y(map.)55 b(Let)36 b Fm(f)p Fo(e)507 1865 y Fk(\013)558 1851 y Fm(g)f Fs(b)s(e)g(a)h(lo)s(cally)e(de\014ned)g(adapted)i(frame)f (relativ)m(e)g(to)i Fo(F)13 b Fs(\()p Fo(M)d Fs(\))36 b(in)e Fl(R)2982 1818 y Fk(n)3029 1851 y Fs(,)j(and)e(let)g Fm(f)p Fo(\022)3500 1818 y Fk(\013)3549 1851 y Fm(g)0 1964 y Fs(denote)d(the)h(corresp)s(onding)c(dual)i(basis.)44 b(W)-8 b(e)33 b(de\014ne)e(a)i(system)f(of)g(lo)s(cal)f(co)s(ordinates) g(\()p Fo(x;)15 b(v)s Fs(\))34 b(in)c Fo(N)3536 1978 y Fk(F)0 2077 y Fs(b)m(y)g(writing)f Fo(v)f Fm(2)d Fo(N)671 2095 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))855 2077 y Fs(as)30 b Fo(v)f Fs(=)c Fo(v)1182 2044 y Fk(a)1224 2077 y Fo(e)1266 2091 y Fk(a)1308 2077 y Fs(\()p Fo(x)p Fs(\).)41 b(Consider)29 b(the)h(lo)s(cal)g(frame)g Fm(f)p Fo(f)2592 2092 y Fk(\014)2639 2077 y Fm(g)c(\032)f Fo(T)13 b(N)2945 2091 y Fk(F)3034 2077 y Fs(de\014ned)29 b(b)m(y)571 2336 y Fo(f)616 2350 y Fk(i)669 2336 y Fs(=)c Fo(e)807 2350 y Fk(i)835 2336 y Fo(;)107 b Fs(1)25 b Fm(\024)g Fo(i)h Fm(\024)f Fo(m)p Fs(;)197 b(and)181 b Fo(f)1961 2350 y Fk(a)2028 2336 y Fs(=)2178 2274 y Fo(@)p 2134 2315 143 4 v 2134 2398 a(@)5 b(v)2234 2372 y Fk(a)2286 2336 y Fo(;)106 b(m)20 b Fs(+)g(1)26 b Fm(\024)f 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Fs(\)\))g(for)f Fo(m)20 b Fs(+)g(1)26 b Fm(\024)f Fo(a;)15 b(b)25 b Fm(\024)g Fo(n)p Fs(.)1152 5328 y Fo(\022)1198 5291 y Fk(a)1239 5328 y Fs(\()p Fo(d\027)1366 5342 y Fk(F)1425 5328 y Fs(\()p Fo(f)1505 5343 y Fk(b)1539 5328 y Fs(\)\))h(=)f Fo(\022)1777 5291 y Fk(a)1818 5173 y Fi( )1941 5267 y Fo(@)p 1900 5308 135 4 v 1900 5391 a(@)5 b(v)2000 5365 y Fk(b)2045 5328 y Fs(\()p Fo(F)34 b Fs(+)20 b Fo(v)s Fs(\))2345 5173 y Fi(!)2417 5328 y Fo(:)p eop %%Page: 7 15 7 14 bop 3848 100 a Fs(7)299 490 y(After)31 b(computing)e(the)i(deriv) -5 b(ativ)m(e)29 b(and)h(simplifying,)d(w)m(e)j(get)1751 711 y Fo(\022)1797 674 y Fk(a)1838 711 y Fs(\()p Fo(d\027)1965 725 y Fk(F)2024 711 y Fs(\()p Fo(f)2104 726 y Fk(b)2139 711 y Fs(\)\))c(=)f Fo(\016)2374 674 y Fk(a)2371 734 y(b)2416 711 y Fo(:)-2142 b Fs(\(2.9\))299 932 y(Using)36 b(\(2)p Fo(:)p Fs(7\))p Fo(;)d Fs(\(2)p Fo(:)p Fs(8\))39 b(and)e(\(2)p Fo(:)p Fs(9\),)k(w)m(e)c(see)h(that)g(the)g(matrix)e (represen)m(tation)h(of)h Fo(\027)3184 946 y Fk(F)3242 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Fk(j)1980 2996 y(i)2017 2968 y Fo(X)2099 2931 y Fk(i)2147 2907 y Fo(@)5 b(F)p 2138 2948 142 4 v 2138 3031 a(@)g(x)2243 3005 y Fk(j)2290 2809 y Fi(\014)2290 2864 y(\014)2290 2918 y(\014)2290 2973 y(\014)2290 3028 y(\014)2320 3086 y Fk(x)2364 2968 y Fo(:)0 3262 y Fs(This)27 b(is)h(called)g(the)h Fp(formal)k(shap)-5 b(e)32 b(op)-5 b(er)g(ator)42 b Fs(of)29 b Fo(j)1747 3218 y Fk(p)1742 3274 y(x)1787 3262 y Fo(F)42 b Fs(along)29 b Fo(v)g Fm(2)24 b Fo(N)2359 3281 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))2513 3262 y Fs(.)40 b(It)29 b(is)e(a)j(direct)e (computation)0 3375 y(to)35 b(sho)m(w)e(that)i(this)e(de\014nition)e (is)i(co)s(ordinate)h(indep)s(enden)m(t.)49 b(It)34 b(follo)m(ws)f (from)h(Remark)f(2.1.1)j(that)0 3488 y(at)e(eac)m(h)h Fo(x)30 b Fm(2)g Fo(M)10 b Fs(,)35 b(the)e(lo)s(cal)g(expression)f(for) h(the)h(shap)s(e)e(op)s(erator)i(of)g Fo(F)46 b Fs(coincides)33 b(with)f(its)h(formal)0 3601 y(analog)40 b Fo(S)353 3615 y Fk(v)393 3601 y Fs(\()p Fo(j)470 3557 y Fk(p)465 3613 y(x)511 3601 y Fo(F)13 b Fs(\).)68 b(In)39 b(particular,)h(if)e(w)m(e)i (let)f Fo(I)48 b Fs(:)40 b Fo(T)1870 3615 y Fk(x)1914 3601 y Fo(M)50 b Fm(!)40 b Fo(T)2236 3615 y Fk(x)2280 3601 y Fo(M)50 b Fs(denote)40 b(the)f(iden)m(tit)m(y)g(map,)j(the)0 3714 y(determinan)m(t)30 b(det\()p Fo(I)e Fm(\000)20 b Fo(S)889 3728 y Fk(v)929 3714 y Fs(\()p Fo(j)1006 3670 y Fk(p)1001 3726 y(x)1047 3714 y Fo(F)13 b Fs(\)\))31 b(is)f(an)g(in)m(v)-5 b(arian)m(tly)29 b(de\014ned)g(smo)s(oth)h (function)f(on)h Fo(J)3067 3670 y Fk(p)3058 3726 y(m)3125 3714 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3372 3681 y Fk(n)3419 3714 y Fs(\).)136 3828 y(W)-8 b(e)32 b(are)f(no)m(w)f(prepared)f(to)i (state)h(our)e(formal)g(de\014nition)e(of)i(\012-regular)g(jets.)0 4022 y Fl(De\014nition)35 b(2.3.1)47 b Fp(We)36 b(say)i(that)g Fo(j)1315 3978 y Fk(p)1310 4034 y(x)1356 4022 y Fo(F)46 b Fm(2)33 b Fo(J)1613 3978 y Fk(p)1604 4034 y(m)1671 4022 y Fs(\()p Fo(M)5 b(;)15 b Ff(R)1918 3989 y Fk(n)1965 4022 y Fs(\))37 b Fp(is)g Fs(\012)p Fp(-)p Fs(regular)f 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Fs(-jets)h(is)f(denoted)h(b)m(y)f Fo(J)1803 5239 y Fk(p)1794 5312 y Fn(\012)1850 5284 y Fs(.)39 b(This)24 b(is)h(an)h(op)s(en)f(submanifold)d(of)k Fo(J)3207 5239 y Fk(p)3198 5295 y(m)3265 5284 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3513 5251 y Fk(n)3559 5284 y Fs(\))0 5396 y(since)30 b(the)g(conditions)f(\(i\))h(to)i(\(iv\))e(ab)s(o)m(v)m (e)h(are)g(op)s(en.)p eop %%Page: 9 17 9 16 bop 3848 100 a Fs(9)299 490 y Fc(2.4)105 b(T)-9 b(ransv)m(erse)35 b(statistics)299 693 y Fs(In)29 b(this)g(section)h(w) m(e)g(de\014ne)f(v)-5 b(arious)29 b(statistical)h(quan)m(tities)f(asso) s(ciated)h(with)f(a)h(probabilit)m(y)d(densit)m(y)299 806 y Fo(p)36 b Fs(and)g(an)g(\012-regular)f(em)m(b)s(edding)g Fo(F)13 b Fs(.)58 b(F)-8 b(or)37 b Fo(x)e Fm(2)g Fo(M)10 b Fs(,)38 b(the)f(restriction)e(of)h Fo(p)g Fs(to)i(\012)3218 824 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))3407 806 y Fs(is)35 b(called)g(the)299 919 y Fp(tr)-5 b(ansverse)35 b(density)c Fs(induced)e(b)m(y)i Fo(p)p Fs(.)42 b(W)-8 b(e)32 b(can)g(do)f(m)m (ultiv)-5 b(ariate)29 b(analysis)h(on)h(\012)3118 937 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))3302 919 y Fs(and)30 b(talk)h(ab)s(out)299 1031 y(the)i Fp(me)-5 b(an)33 b Fs(and)f(the)h Fp(c)-5 b(ovarianc)g(e)36 b(matrix)e Fs(of)e(the)h (transv)m(erse)g(probabilit)m(y)d(densit)m(y)-8 b(.)47 b(These)32 b(concepts)299 1144 y(arise)e(naturally)e(as)j(w)m(e)g(use)f (a)h(system)f(of)h(normal)e(co)s(ordinates)h(of)h Fo(F)43 b Fs(in)29 b(computations.)435 1257 y(W)-8 b(e)35 b(assume)e(that)h (for)g(almost)f(ev)m(ery)i Fo(x)30 b Fm(2)g Fo(M)44 b Fs(the)34 b(set)g Fo(supp)p Fs(\()p Fo(p)p Fs(\))22 b Fm(\\)g Fs(\012)2879 1276 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))3065 1257 y Fs(has)33 b(p)s(ositiv)m(e)g(measure)299 1370 y(as)d(a)h(subset)f(of)h(the)f(disk)f(\012)1281 1389 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))1434 1370 y Fs(.)299 1555 y Fl(De\014nition)35 b(2.4.1)46 b Fp(The)41 b Fo(k)s Fp(-th)48 b Fs(transv)m(erse)40 b(momen)m(t)h Fp(is)g(the)g(map)h Fs(\003)2785 1570 y Fk(k)2867 1555 y Fs(:)e Fo(M)50 b Fm(!)40 b Fo(H)7 b(om)p Fs(\()p Ff(R)3521 1522 y Fk(n)3594 1555 y Fm(\002)26 b(\001)15 b(\001)g(\001)27 b(\002)299 1668 y Ff(R)377 1635 y Fk(n)424 1668 y Fo(;)15 b Ff(R)q Fs(\))32 b Fp(de\014ne)-5 b(d)34 b(by)1041 1912 y Fs(\003)1104 1926 y Fn(0)1144 1912 y Fs(\()p Fo(x)p Fs(\))83 b(=)1503 1788 y Fi(Z)1553 1994 y Fn(\012)1604 2008 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))1757 1912 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s Fs(\))p Fo(dv)s(;)853 b(k)29 b Fs(=)c(0)p Fo(;)700 2178 y Fs(\003)763 2193 y Fk(k)806 2178 y Fs(\()p Fo(X)916 2192 y Fn(1)956 2178 y Fo(;)15 b(:::;)g(X)1186 2193 y Fk(k)1231 2178 y Fs(\))83 b(=)1503 2054 y Fi(Z)1553 2260 y Fn(\012)1604 2274 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))1742 2178 y Fm(h)p Fo(X)1852 2192 y Fn(1)1892 2178 y Fo(;)15 b(v)s Fm(i)g(\001)g(\001)g(\001)j(h)p Fo(X)2262 2193 y Fk(k)2305 2178 y Fo(;)d(v)s Fm(i)p Fo(p)p Fs(\()p Fo(F)e Fs(\()p Fo(x)p Fs(\))22 b(+)e Fo(v)s Fs(\))p Fo(dv)s(;)235 b(k)29 b(>)c Fs(0)p Fo(;)299 2447 y Fp(for)33 b Fo(X)519 2461 y Fn(1)559 2447 y Fo(;)15 b(:::;)g(X)789 2462 y Fk(k)859 2447 y Fm(2)24 b Ff(R)1023 2414 y Fk(n)1070 2447 y Fp(.)299 2657 y Fl(De\014nition)35 b(2.4.2)46 b Fp(The)40 b Fs(mean)33 b Fp(of)g(the)g(tr)-5 b(ansverse)34 b(density)f(is)g(the)g(map)1609 2858 y Fo(\026)1664 2873 y Fe(?)1748 2858 y Fs(:)25 b Fo(M)35 b Fm(!)26 b Fo(H)7 b(om)p Fs(\()p Ff(R)2358 2820 y Fk(n)2405 2858 y Fo(;)15 b Ff(R)p Fs(\))p Fo(;)299 3059 y Fp(de\014ne)-5 b(d)34 b(by)1841 3261 y Fo(\026)1896 3276 y Fe(?)1980 3261 y Fs(=)25 b(\003)2139 3275 y Fn(1)2178 3261 y Fo(=)p Fs(\003)2286 3275 y Fn(0)2326 3261 y Fo(:)299 3462 y Fp(The)33 b Fs(co)m(v)-5 b(ariance)24 b(tensor)i Fp(of)g(the)g(tr)-5 b(ansverse)28 b(density)e(is)g(the)g(symmetric)h(map)g Fo(\033)3079 3429 y Fn(2)3076 3490 y Fe(?)3160 3462 y Fs(:)f Fo(M)35 b Fm(!)25 b Fo(H)7 b(om)p Fs(\()p Ff(R)3771 3429 y Fk(n)3823 3462 y Fm(\002)299 3575 y Ff(R)377 3542 y Fk(n)424 3575 y Fo(;)15 b Ff(R)q Fs(\))32 b Fp(de\014ne)-5 b(d)34 b(by)1617 3776 y Fo(\033)1672 3739 y Fn(2)1669 3799 y Fe(?)1754 3776 y Fs(=)25 b(\003)1913 3790 y Fn(2)1952 3776 y Fo(=)p Fs(\003)2060 3790 y Fn(0)2120 3776 y Fm(\000)20 b Fo(\026)2266 3791 y Fe(?)2345 3776 y Fm(\012)g Fo(\026)2491 3791 y Fe(?)2550 3776 y Fo(:)299 3986 y Fl(Example)34 b(2.4.3)46 b Fp(R)-5 b(adial)36 b(distributions)f(r)-5 b(elative)35 b(to)f(a)g(manifold)i Fo(M)10 b Fp(.)46 b Fs(Let)32 b(\012)f(b)s(e)g(a) h(tubular)e(neigh-)299 4099 y(b)s(orho)s(o)s(d)36 b(of)i(a)g(manifold)e Fo(M)48 b Fs(with)37 b(constan)m(t)i(radius,)f(and)f(let)h Fo(p)g Fs(b)s(e)f(a)i(probabilit)m(y)c(densit)m(y)i(on)h(\012)299 4212 y(dep)s(ending)d(only)h(on)i(the)f(distance)h(to)g Fo(M)10 b Fs(.)62 b(W)-8 b(e)38 b(call)f Fo(p)g Fs(a)h Fp(r)-5 b(adial)41 b(density)f(r)-5 b(elative)40 b(to)k Fo(M)10 b Fs(.)62 b(In)37 b(this)299 4325 y(case,)c(the)g(transv)m (erse)f(mean)g Fo(\026)1408 4340 y Fe(?)1498 4325 y Fs(v)-5 b(anishes)31 b(iden)m(tically)f(and)h(the)h(transv)m(erse)h(co)m(v)-5 b(ariance)33 b(tensor)f Fo(\033)3838 4292 y Fn(2)3835 4352 y Fe(?)299 4437 y Fs(is)27 b(a)h(constan)m(t)h(m)m(ultiple)c(of)j (the)g(Euclidean)e(scalar)h(pro)s(duct)g(restricted)g(to)i(the)f(disks) e(\012)3399 4456 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))3552 4437 y Fo(;)30 b(x)c Fm(2)e Fo(M)10 b Fs(.)299 4550 y(T)-8 b(o)35 b(see)g(this,)g(write)f Fo(v)i Fm(2)c Fs(\012)1272 4569 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))1459 4550 y Fs(in)33 b(terms)i(of)g(the)g(orthonormal)f(frame)g Fm(f)p Fo(e)2967 4564 y Fn(1+)p Fk(m)3125 4550 y Fo(;)15 b(:::;)g(e)3322 4564 y Fk(n)3371 4550 y Fm(g)32 b(\032)h Fo(N)3625 4569 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))3812 4550 y Fs(as)299 4663 y Fo(v)30 b Fs(=)c Fo(v)517 4630 y Fk(a)559 4663 y Fo(e)601 4677 y Fk(a)643 4663 y Fs(.)42 b(Since)31 b(\012)f(is)g(a)i(radially)d(symmetric)h(tubular)f(neigh)m(b)s(orho)s (o)s(d)g(of)i Fo(M)41 b Fs(and)31 b Fo(p)f Fs(dep)s(ends)g(only)299 4776 y(on)g(the)h(distance)f(to)h Fo(M)10 b Fs(,)31 b(all)e(of)h(the)h (transv)m(erse)g(in)m(tegrals)505 4891 y Fi(Z)556 5097 y Fn(\012)607 5111 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))760 5014 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s Fs(\))p Fo(dv)s(;)1545 4891 y Fi(Z)1596 5097 y Fn(\012)1647 5111 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))1799 5014 y Fo(v)1846 4977 y Fk(a)1888 5014 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s Fs(\))p Fo(dv)s(;)2674 4891 y Fi(Z)2724 5097 y Fn(\012)2775 5111 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))2928 5014 y Fo(v)2975 4977 y Fk(a)3017 5014 y Fo(v)3064 4977 y Fk(b)3099 5014 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s Fs(\))p Fo(dv)s(;)299 5284 y Fs(are)38 b(constan)m(t.)64 b(In)37 b(particular,)h(those)h(whose)e(in)m(tegrand)g(dep)s(ends)f(on) i(o)s(dd)f(p)s(o)m(w)m(ers)g(of)h Fo(v)3538 5251 y Fk(a)3618 5284 y Fs(v)-5 b(anish.)299 5396 y(T)d(o)39 b(c)m(hec)m(k)h(this,)g(c)m (hange)f(to)h(p)s(olar)d(co)s(ordinates)h Fo(v)k Fs(=)d Fo(r)s(\022)s Fs(,)h Fo(r)h Fm(\025)e Fs(0,)i Fo(\022)g Fm(2)d Fo(S)2967 5363 y Fk(n)p Fe(\000)p Fn(2)3105 5396 y Fs(,)i(to)g(compute)e(them.)p eop %%Page: 10 18 10 17 bop 0 100 a Fs(10)0 490 y(Since)27 b Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(x)p Fs(\))k(+)f Fo(v)s Fs(\))29 b(is)f(a)h(function)e(of)h Fo(r)j Fs(only)-8 b(,)29 b(in)m(tegration)f (o)m(v)m(er)i Fo(S)2345 457 y Fk(n)p Fe(\000)p Fn(2)2510 490 y Fs(v)-5 b(anishes)27 b(b)s(ecause)i(the)f(v)-5 b(alues)0 603 y(of)35 b Fo(v)155 570 y Fk(a)232 603 y Fs(tak)m(en)h(on)f(one)g(hemisphere)e(are)i(exactly)h(canceled)f(b)m(y) g(the)g(v)-5 b(alues)34 b(tak)m(en)i(on)f(the)g(an)m(tip)s(o)s(dal)0 716 y(hemisphere.)k(Therefore,)30 b(the)g(transv)m(erse)h(mean)f Fo(\026)1832 731 y Fe(?)1921 716 y Fs(v)-5 b(anishes)29 b(as)i(w)m(ell)e(as)h(the)h(o\013-diagonal)f(en)m(tries)0 829 y(of)j(the)g(matrix)g(represen)m(tation)g(of)g Fo(\033)1322 796 y Fn(2)1319 856 y Fe(?)1411 829 y Fs(with)e(resp)s(ect)i(to)h(the)f (basis)f Fm(f)p Fo(e)2520 843 y Fn(1+)p Fk(m)2677 829 y Fo(;)15 b(:::;)g(e)2874 843 y Fk(n)2923 829 y Fm(g)p Fs(.)49 b(Since)32 b(\012)3348 847 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))3534 829 y Fs(is)0 941 y(homothetic)37 b(to)h(the)f(unit)e (disk)g Fo(D)1234 908 y Fk(n)p Fe(\000)p Fk(m)1435 941 y Fs(and)h Fo(p)h Fs(dep)s(ends)d(only)i(on)h Fm(j)p Fo(v)s Fm(j)p Fs(,)i(all)d(of)h(the)g(diagonal)f(en)m(tries)0 1054 y(\()p Fo(\033)90 1021 y Fn(2)87 1082 y Fe(?)146 1054 y Fs(\))181 1021 y Fk(aa)283 1054 y Fs(are)22 b(the)g(same.)39 b(Consequen)m(tly)21 b Fo(\033)1445 1021 y Fn(2)1442 1082 y Fe(?)1523 1054 y Fs(is)g(a)h(constan)m(t)h(m)m(ultiple)d(of)i (the)g(Euclidean)e(scalar)i(pro)s(duct)0 1167 y(restricted)30 b(to)h(\012)582 1186 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))735 1167 y Fs(.)p eop %%Page: 11 19 11 18 bop 1882 805 a Fq(Chapter)34 b(3)443 1005 y Fj(EXTREMAL)i(PR)m (OPER)-9 b(TIES)34 b(OF)k(THE)f(DIST)-9 b(ANCE)35 b(FUNCTIONAL)435 1284 y Fs(Recall)c(that)h(for)e(an)h Fo(m)p Fs(-dimensional)e(compact)j (orien)m(table)f(Riemannian)e(manifold)g Fo(M)41 b Fs(without)299 1396 y(b)s(oundary)28 b(and)i(an)h(\012-regular)e(em)m(b)s(edding)g Fo(F)39 b Fs(:)26 b Fo(M)35 b Fm(!)26 b Fl(R)2342 1363 y Fk(n)2389 1396 y Fs(,)31 b(the)g(distance)f(functional)f(is)g(one)i (half)e(of)299 1509 y(the)h(exp)s(ected)h(v)-5 b(alue)30 b(of)h(the)f(square)g(of)h(distance)f(to)h Fo(F)13 b Fs(\()p Fo(M)d Fs(\),)1330 1738 y Fo(E)1397 1753 y Fk(d)1438 1738 y Fs([)p Fo(F)j Fs(])26 b(=)1691 1676 y(1)p 1691 1717 46 4 v 1691 1800 a(2)1761 1614 y Fi(Z)1812 1820 y Fn(\012)1882 1738 y Fm(j)p Fl(y)21 b Fm(\000)e Fo(F)13 b Fs(\()p Fo(\025)2232 1752 y Fk(F)2292 1738 y Fs(\()p Fl(y)p Fs(\)\))p Fm(j)2478 1696 y Fn(2)2548 1738 y Fo(p)p Fs(\()p Fl(y)p Fs(\))i Fo(d)p Fl(y)q Fo(;)299 1965 y Fs(where)40 b Fo(\025)625 1979 y Fk(F)724 1965 y Fs(is)f(the)i(pro)5 b(jection)40 b(index)f(and)h Fo(p)g Fs(is)g(the)h(probabilit)m(y)d (densit)m(y)-8 b(.)71 b(As)40 b(w)m(e)h(men)m(tioned)f(in)299 2078 y(the)30 b(in)m(tro)s(duction,)f(this)g(functional)f(do)s(es)i (not)g(ha)m(v)m(e)i(lo)s(cal)d(minima)f(within)f(the)k(class)f(of)g (smo)s(oth)g(\012-)299 2191 y(regular)d(em)m(b)s(eddings.)39 b(In)27 b(this)g(c)m(hapter)i(w)m(e)g(pro)m(v)m(e)g(this)e(fact)i(b)m (y)g(studying)d(the)j(extremal)f(prop)s(erties)299 2304 y(of)i Fo(E)469 2319 y Fk(d)510 2304 y Fs(.)299 2551 y Fc(3.1)105 b(F)-9 b(unction)36 b(spaces)299 2754 y Fs(Up)22 b(to)h(no)m(w)f(the)h(functional)e Fo(E)1359 2769 y Fk(d)1422 2754 y Fs(has)h(b)s(een)g(de\014ned)f(for)h(smo)s(oth) g(em)m(b)s(eddings.)36 b(T)-8 b(o)23 b(study)f(its)g(extremal)299 2867 y(prop)s(erties)28 b(w)m(e)i(consider)e(it)h(as)h(a)g(functional)e (de\014ned)g(on)h(a)h(Banac)m(h)h(space.)41 b(Since)28 b(these)i(prop)s(erties)299 2980 y(can)37 b(b)s(e)g(form)m(ulated)f(in) g(terms)h(of)g(di\013eren)m(tial)f(equations,)j(it)d(is)h(con)m(v)m (enien)m(t)h(to)g(consider)e(Sob)s(olev)299 3093 y(spaces)31 b(for)f(the)g(domain)g(of)g Fo(E)1367 3108 y Fk(d)1408 3093 y Fs(.)435 3294 y(F)-8 b(or)29 b(an)f Fo(m)p Fs(-dimensional)e (compact)j(Riemannian)d(manifold)g Fo(M)39 b Fs(without)27 b(a)h(b)s(oundary)-8 b(,)28 b(w)m(e)g(de\014ne)299 3407 y(the)34 b(Sob)s(olev)f(spaces)h(\000)1140 3374 y Fk(k)1182 3407 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1429 3374 y Fk(n)1476 3407 y Fs(\))p Fo(;)31 b(k)j Fs(=)d(0)p Fo(;)15 b Fs(1)p Fo(;)g(:::)p Fs(,)37 b(as)d(follo)m(ws.)50 b(Let)34 b Fm(f)p Fo(U)2789 3421 y Fk(i)2818 3407 y Fm(g)2863 3421 y Fk(i)p Fe(2)p Fk(I)3008 3407 y Fs(b)s(e)f(a)i(\014nite)d(co)m(v)m(er) k(of)e Fo(M)299 3520 y Fs(b)m(y)g(lo)s(cal)f(c)m(harts)h (di\013eomorphic)e(to)j(op)s(en)e(b)s(ounded)f(subsets)h(of)h Fl(R)2710 3487 y Fk(m)2776 3520 y Fs(.)52 b(Let)34 b Fm(f)p Fo(')3123 3534 y Fk(i)3152 3520 y Fm(g)g Fs(b)s(e)g(a)g (partition)e(of)299 3632 y(unit)m(y)d(sub)s(ordinate)f(to)k Fm(f)p Fo(U)1253 3646 y Fk(i)1281 3632 y Fm(g)p Fs(.)41 b(F)-8 b(or)32 b Fo(F)38 b Fm(2)25 b Fo(C)1810 3599 y Fe(1)1884 3632 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2131 3599 y Fk(n)2178 3632 y Fs(\),)31 b(w)m(e)g(let)1430 3861 y Fm(jj)p Fo(')1539 3875 y Fk(i)1569 3861 y Fo(F)13 b Fm(jj)1690 3823 y Fn(2)1690 3884 y Fk(k)1758 3861 y Fs(=)1877 3775 y Fi(X)1854 3976 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)2047 3737 y Fi(Z)2098 3943 y Fk(U)2146 3953 y Fg(i)2191 3861 y Fm(j)p Fo(D)2294 3823 y Fk(\013)2344 3861 y Fs(\()p Fo(')2438 3875 y Fk(i)2467 3861 y Fo(F)g Fs(\))p Fm(j)2598 3823 y Fn(2)2638 3861 y Fo(dx;)299 4147 y Fs(b)s(e)33 b(the)h(square)f(of)h(the)g Fo(k)s Fs(-Sob)s(olev)g(norm)f(of)g(the)h (v)m(ector)i(v)-5 b(alued)32 b(function)h Fo(')3056 4161 y Fk(i)3085 4147 y Fo(F)43 b Fs(:)32 b Fo(U)3305 4161 y Fk(i)3364 4147 y Fm(!)f Fl(R)3564 4114 y Fk(n)3611 4147 y Fs(.)51 b(Then)299 4260 y(\000)356 4227 y Fk(k)398 4260 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)646 4227 y Fk(n)693 4260 y Fs(\))30 b(is)g(the)g(completion)g(of)g Fo(C)1646 4227 y Fe(1)1721 4260 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1968 4227 y Fk(n)2015 4260 y Fs(\))30 b(with)f(resp)s(ect)i(to)g(the)g(norm) 1557 4547 y Fm(jj)p Fo(F)13 b Fm(jj)1728 4562 y Fk(k)1797 4547 y Fs(=)1893 4391 y Fi( )1980 4461 y(X)1992 4657 y Fk(i)p Fe(2)p Fk(I)2126 4547 y Fm(jj)p Fo(')2235 4561 y Fk(i)2264 4547 y Fo(F)g Fm(jj)2385 4509 y Fn(2)2385 4570 y Fk(k)2429 4391 y Fi(!)2501 4414 y Fn(1)p Fk(=)p Fn(2)2611 4547 y Fo(:)299 4832 y Fs(F)-8 b(or)44 b Fo(k)49 b Fm(\025)d Fs(0,)h(let)c Fo(C)1021 4799 y Fk(k)1063 4832 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1310 4799 y Fk(n)1357 4832 y Fs(\))44 b(b)s(e)e(the)h(Banac)m(h)i(space)e(of)g Fo(C)2522 4799 y Fk(k)2607 4832 y Fs(functions)f(with)g(the)h(norm)f (of)h(the)299 4945 y(supreme.)51 b(Sob)s(olev's)33 b(Lemma)h (establishes)f(that)i(the)f(inclusion)d(\000)2692 4912 y Fk(k)r Fn(+)p Fk(k)2823 4921 y Fb(0)2861 4945 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3108 4912 y Fk(n)3155 4945 y Fs(\))32 b Fo(,)-15 b Fm(!)32 b Fo(C)3427 4912 y Fk(k)3469 4945 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3716 4912 y Fk(n)3763 4945 y Fs(\))35 b(is)299 5058 y(con)m(tin)m(uous,)h(where)e Fo(k)1099 5072 y Fn(0)1171 5058 y Fs(=)e(1)24 b(+)f([)1472 5022 y Fk(m)p Fe(\000)p Fn(1)p 1472 5037 153 4 v 1531 5089 a(2)1634 5058 y Fs(],)37 b(and)d([)1937 5022 y Fk(m)p Fe(\000)p Fn(1)p 1937 5037 V 1996 5089 a(2)2100 5058 y Fs(])g(denotes)i(the)f(least)g(in)m(teger)g(bigger)f(or)h(equal)f(to) 299 5171 y(\()p Fo(m)20 b Fm(\000)g Fs(1\))p Fo(=)p Fs(2.)435 5284 y(W)-8 b(e)30 b(extend)f(the)g(class)f(of)h(\012-regular)f(em)m(b) s(eddings)f Fm(E)2297 5298 y Fn(\012)2380 5284 y Fs(to)j(include)c (those)j(em)m(b)s(eddings)e(in)h(\000)3693 5251 y Fk(k)3735 5284 y Fs(\()p Fo(M)10 b Fs(,)299 5396 y Fl(R)377 5363 y Fk(n)424 5396 y Fs(\))42 b(satisfying)e(conditions)f(\(i\))i(through) g(\(iv\))g(of)g(De\014nition)f(1.0.1.)75 b(W)-8 b(e)42 b(denote)g(this)e(subset)g(of)p eop %%Page: 12 20 12 19 bop 0 100 a Fs(12)0 490 y(\000)57 457 y Fk(k)99 490 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)347 457 y Fk(n)394 490 y Fs(\))31 b(b)m(y)h Fm(E)644 457 y Fk(k)636 517 y Fn(\012)691 490 y Fs(.)44 b(F)-8 b(or)32 b Fo(k)e Fm(\025)d Fs(2)21 b(+)g Fo(k)1304 504 y Fn(0)1344 490 y Fs(,)32 b(this)e(is)g(an)i(op)s(en)f(subset)f(of)i(\000)2461 457 y Fk(k)2504 490 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2751 457 y Fk(n)2798 490 y Fs(\).)44 b(This)30 b(follo)m(ws)g(from)0 603 y(Sob)s(olev's)37 b(Lemma)i(and)e(from)h(the)h(fact)g(that)g(the)f (subset)g(of)g(\012-regular)g(t)m(w)m(o-jets)i Fo(J)3087 570 y Fn(2)3078 630 y(\012)3172 603 y Fs(is)d(an)h(op)s(en)0 716 y(submanifold)21 b(of)k Fo(J)664 683 y Fn(2)704 716 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)951 683 y Fk(n)998 716 y Fs(\);)27 b(i.e.,)f(an)m(y)f(em)m(b)s(edding)e Fo(F)38 b Fm(2)25 b(E)2110 683 y Fk(k)2102 743 y Fn(\012)2181 716 y Fs(has)g(a)g(neigh)m(b)s(orho)s(o)s(d)d Fo(U)35 b Fm(\032)25 b Fs(\000)3223 683 y Fk(k)3265 716 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3513 683 y Fk(n)3559 716 y Fs(\))0 829 y(suc)m(h)33 b(that)h(for)g(all)700 806 y(~)680 829 y Fo(F)44 b Fm(2)30 b Fo(U)43 b Fs(the)34 b(di\013erence)f(b)s(et)m(w)m (een)h(the)g(T)-8 b(a)m(ylor)33 b(series)g(expansion)f(up)h(t)m(w)m(o)i (second)0 941 y(order)30 b(of)g Fo(F)44 b Fs(and)640 918 y(~)620 941 y Fo(F)f Fs(is)29 b(so)i(small)e(that)1379 918 y(~)1359 941 y Fo(F)43 b Fs(also)31 b(satis\014es)e(condition)g 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b(e)36 b(de\014ne)d(some)i(quan)m (tities)e(\014rst.)51 b(F)-8 b(or)35 b Fo(V)52 b Fm(2)31 b Fo(C)2040 1794 y Fe(1)2114 1827 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2361 1794 y Fk(n)2408 1827 y Fs(\),)36 b(let)e Fo(\016)2682 1794 y Fn(2)2722 1827 y Fo(V)52 b Fs(:)32 b Fo(M)41 b Fm(!)32 b Fo(T)13 b(J)3261 1794 y Fn(2)3252 1854 y(\012)3341 1827 y Fs(b)s(e)34 b(the)0 1940 y(map)c(de\014ned)f (in)g(lo)s(cal)h(co)s(ordinates)g(b)m(y)g(the)h(form)m(ula)1177 2220 y Fo(\016)1220 2183 y Fn(2)1261 2220 y Fo(V)45 b Fs(=)1503 2107 y Fn(2)1455 2134 y Fi(X)1459 2328 y Fk(r)r Fn(=1)1741 2159 y Fo(@)1794 2126 y Fk(r)1832 2159 y Fo(V)1905 2126 y Fk(\013)p 1612 2199 473 4 v 1612 2283 a Fo(@)5 b(x)1717 2256 y Fk(i)1741 2265 y Fb(1)1795 2283 y Fm(\001)15 b(\001)g(\001)h Fo(@)5 b(x)2021 2256 y Fk(i)2045 2264 y Fg(r)2216 2159 y Fo(@)p 2104 2199 279 4 v 2104 2283 a(@)g(y)2205 2251 y Fk(\013)2202 2310 y(i)2226 2319 y Fb(1)2261 2310 y Fe(\001\001\001)n Fk(i)2343 2318 y Fg(r)2392 2220 y Fo(:)-2417 b Fs(\(3.1\))0 2497 y(F)-8 b(or)31 b(j=0,)g(1,...,)h(l;)d(let)978 2829 y Fo(d)1025 2791 y Fn(\()p Fk(j)t Fn(\))1117 2829 y Fo(g)g Fs(:)c Fo(J)1298 2791 y Fn(2)1289 2851 y(\012)1370 2829 y Fm(!)g Fo(H)7 b(om)p Fs(\()1914 2664 y Fk(j)94 b(times)1728 2723 y Fi(z)p 1769 2723 254 11 v 254 w(}|)p 2105 2723 V 254 w({)1728 2829 y Fo(T)13 b(J)1853 2791 y Fn(2)1844 2851 y(\012)1920 2829 y Fm(\002)20 b(\001)15 b(\001)g(\001)21 b(\002)f Fo(T)13 b(J)2353 2791 y Fn(2)2344 2851 y(\012)2399 2829 y Fo(;)i Fl(R)2518 2791 y Fk(q)2556 2829 y Fs(\))p Fo(;)0 3030 y Fs(b)s(e)30 b(the)g Fo(j)5 b Fs(-th)31 b(total)h(deriv)-5 b(ativ)m(e)30 b(of)g Fo(g)s Fs(.)41 b(W)-8 b(e)32 b(de\014ne)e(the)g(map)361 3231 y Fo(C)433 3194 y Fn(\()p Fk(j)t Fn(\))549 3231 y Fs(:)c Fm(E)648 3245 y Fn(\012)723 3231 y Fm(\002)20 b Fo(C)886 3194 y Fk(k)r Fn(+)p Fk(l)q Fn(+1)1095 3231 y Fs(\()p Fo(J)1189 3194 y Fn(2)1180 3254 y(\012)1236 3231 y Fo(;)15 b Fl(R)1354 3194 y Fk(q)1393 3231 y Fs(\))25 b Fm(!)1088 3492 y Fo(H)7 b(om)p Fs(\()1820 3340 y Fk(j)94 b(times)1330 3399 y Fi(z)p 1371 3399 559 11 v 559 w(}|)p 2012 3399 V 559 w({)1330 3492 y Fo(C)1402 3455 y Fe(1)1477 3492 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1724 3455 y Fk(n)1771 3492 y Fs(\))20 b Fm(\002)g(\001)15 b(\001)g(\001)21 b(\002)f Fo(C)2206 3455 y Fe(1)2281 3492 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2528 3455 y Fk(n)2575 3492 y Fs(\))p Fo(;)g(C)2722 3455 y Fk(k)r Fn(+)p Fk(l)q Fn(+1)p Fe(\000)p Fk(j)3019 3492 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3266 3455 y Fk(q)3304 3492 y Fs(\)\))p Fo(;)0 3693 y Fs(giv)m(en)30 b(b)m(y)251 3821 y Fi(\000)292 3895 y Fo(C)364 3857 y Fn(\()p Fk(j)t Fn(\))455 3895 y Fs(\()p Fo(F)s(;)15 b(g)s Fs(\))672 3821 y Fi(\001)715 3895 y Fs(\()p Fo(V)803 3909 y Fn(1)843 3895 y Fo(;)g(:::;)g(V)1051 3909 y Fk(j)1089 3895 y Fs(\))1124 3909 y Fk(x)1194 3895 y Fs(=)1290 3821 y Fi(\000)1332 3895 y Fo(d)1379 3857 y Fn(\()p Fk(j)t Fn(\))1470 3895 y Fo(g)s Fs(\()p Fo(j)1593 3857 y Fn(2)1588 3917 y Fk(x)1634 3895 y Fo(F)e Fs(\))1740 3821 y Fi(\001)1782 3895 y Fs(\()p Fo(\016)1860 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b(\014xed)f Fo(g)s Fs(,)h(w)m(e)g(sho)m(w)f(that)h Fo(C)1340 5363 y Fn(\()p Fk(j)t Fn(\))1431 5396 y Fs(\()p Fm(\001)p Fo(;)15 b(g)s Fs(\))33 b(is)c(the)h Fo(j)5 b Fs(-th)32 b(deriv)-5 b(ativ)m(e)30 b(of)g Fo(C)7 b Fs(\()p Fm(\001)p Fo(;)15 b(g)s Fs(\).)p eop %%Page: 13 21 13 20 bop 3803 100 a Fs(13)365 490 y(\(3\))46 b(Finally)-8 b(,)29 b(w)m(e)i(sho)m(w)f(that)h Fo(C)37 b Fs(is)29 b(a)i Fo(C)1750 457 y Fk(l)1806 490 y Fs(map.)299 702 y(F)-8 b(or)41 b(the)g(pro)s(of)f(of)g(the)h(lemma,)i(a)e(form)m(ula)f (to)h(compute)g(the)f(deriv)-5 b(ativ)m(es)40 b(of)h Fo(C)3243 669 y Fn(\()p Fk(j)t Fn(\))3334 702 y Fs(\()p Fo(F)s(;)15 b(g)s Fs(\))42 b(in)d(lo)s(cal)299 815 y(co)s(ordinates)32 b(is)f(needed.)46 b(Let)33 b(\()p Fo(U)e Fm(\002)21 b Fl(R)1685 782 y Fk(n)1732 815 y Fs(;)15 b Fo(x)1824 782 y Fk(i)1852 815 y Fo(;)g(y)1940 782 y Fk(\015)1985 815 y Fs(\))32 b(b)s(e)g(a)h(lo)s(cal)e(co)s(ordinate)h(system)g(on)g Fo(M)g Fm(\002)21 b Fl(R)3642 782 y Fk(n)3689 815 y Fs(,)33 b(and)299 928 y(let)28 b(\()p Fo(W)13 b Fs(;)i Fo(x)654 895 y Fk(i)682 928 y Fo(;)g(y)770 895 y Fk(\015)815 928 y Fo(;)g(y)903 883 y Fk(\015)900 955 y(i)948 928 y Fo(;)g(:::;)g(y)1151 883 y Fk(\015)1148 955 y(i)1172 964 y Fb(1)1207 955 y Fe(\001\001\001)o Fk(i)1290 964 y Fb(2)1329 928 y Fs(\))28 b(b)s(e)g(the)g(induced)e(lo)s(cal)i(co)s(ordinate)g(system)g(in)f Fo(J)3124 895 y Fn(2)3115 954 y(\012)3170 928 y Fs(.)40 b(F)-8 b(or)29 b(a)f(m)m(ultiindex)299 1040 y Fo(\013)j Fs(=)g(\()p Fo(a)573 1054 y Fn(1)613 1040 y Fo(;)15 b(:::;)g(a)816 1054 y Fk(m)884 1040 y Fs(\))p Fo(;)31 b(a)1023 1054 y Fk(i)1082 1040 y Fm(2)g Fl(N)p Fs(,)j(let)g Fm(j)p Fo(\013)p Fm(j)e Fs(=)1691 972 y Fi(P)1787 999 y Fk(m)1787 1067 y(i)p Fn(=1)1921 1040 y Fo(a)1969 1054 y Fk(i)1997 1040 y Fs(.)51 b(Denote)35 b(the)f Fo(\013)p Fs(-deriv)-5 b(ativ)m(e)34 b(of)g(a)g(smo)s(oth)g(function)299 1153 y Fo(f)g Fm(2)25 b Fo(C)536 1120 y Fe(1)610 1153 y Fs(\()p Fo(U)10 b Fs(\))31 b(b)m(y)1537 1419 y Fo(D)1615 1382 y Fk(\013)1664 1419 y Fo(f)k Fs(=)2137 1358 y Fo(@)2190 1325 y Fe(j)p Fk(\013)p Fe(j)2279 1358 y Fo(f)p 1850 1398 771 4 v 1850 1482 a(@)5 b Fs(\()p Fo(x)1990 1455 y Fn(1)2030 1482 y Fs(\))2065 1455 y Fk(a)2102 1464 y Fb(1)2156 1482 y Fm(\001)15 b(\001)g(\001)i Fo(@)5 b Fs(\()p Fo(x)2418 1455 y Fk(m)2485 1482 y Fs(\))2520 1455 y Fk(a)2557 1463 y Fg(m)2630 1419 y Fo(:)299 1683 y Fs(F)-8 b(or)31 b(an)f(em)m(b)s(edding)f Fo(F)38 b Fm(2)25 b(E)1278 1697 y Fn(\012)1363 1683 y Fs(and)30 b(a)h(function)e Fo(g)g Fm(2)c Fo(C)2202 1650 y Fk(k)r Fn(+)p Fk(l)q Fn(+1)2411 1683 y Fs(\()p Fo(J)2505 1650 y Fn(2)2496 1709 y(\012)2552 1683 y Fo(;)15 b Fl(R)2671 1650 y Fk(q)2709 1683 y Fs(\),)30 b(w)m(e)h(compute)854 1818 y Fi(X)830 2019 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)1023 1904 y Fo(D)1101 1866 y Fk(\013)1151 1830 y Fi(\000)1192 1904 y Fs(\()p Fo(C)1299 1866 y Fn(\()p Fk(j)t Fn(\))1390 1904 y Fs(\()p Fo(F)s(;)15 b(g)s Fs(\)\)\()p Fo(V)1730 1918 y Fn(1)1772 1904 y Fo(;)g(:::;)g(V)1980 1918 y Fk(j)2019 1904 y Fs(\))2054 1918 y Fk(x)2098 1830 y Fi(\001)677 2259 y Fs(=)830 2137 y Fn([)p Fk(k)r(=)p Fn(2])p Fe(\000)p Fn(1)884 2173 y Fi(X)892 2368 y Fk(i)p Fn(=0)1400 2173 y Fi(X)1084 2374 y Fe(j)p Fk(\013)1149 2383 y Fb(0)1183 2374 y Fn(+)p Fk(\013)1283 2383 y Fb(1)1318 2374 y Fn(+)p Fe(\001\001\001)o Fn(+)p Fk(\013)1532 2384 y Fg(j)1565 2374 y Fe(j\024)p Fk(k)r Fn(+2)p Fe(\000)p Fk(i)1862 2259 y Fo(g)1905 2273 y Fk(i;\013)1994 2282 y Fb(0)2029 2273 y Fe(\001\001\001)o Fk(\013)2133 2283 y Fg(j)2170 2259 y Fs(\()p Fo(j)2247 2221 y Fk(i)p Fn(+2)2242 2281 y Fk(x)2366 2259 y Fo(F)e Fs(\)\()p Fo(D)2585 2221 y Fk(\013)2630 2230 y Fb(0)2670 2259 y Fo(F)s(;)i(D)2849 2221 y Fk(\013)2894 2230 y Fb(1)2933 2259 y Fo(V)2986 2273 y Fn(1)3025 2259 y Fo(;)g(:::;)g(D)3258 2221 y Fk(\013)3303 2231 y Fg(j)3342 2259 y Fo(V)3395 2273 y Fk(j)3431 2259 y Fs(\);)-3192 b(\(3.3\))2648 2504 y Fo(V)2701 2518 y Fn(1)2741 2504 y Fo(;)15 b(:::;)g(V)2949 2518 y Fk(j)3012 2504 y Fm(2)25 b Fo(C)3170 2467 y Fe(1)3244 2504 y Fs(\()p Fo(M)5 b(;)15 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b(the)h(de\014nition)e(\(3)p Fo(:)p Fs(2\).)43 b(F)-8 b(or)31 b Fo(j)g Fs(=)25 b(1,)31 b(w)m(e)g(get)1204 3284 y Fi(\000)1246 3357 y Fo(C)1318 3320 y Fn(\(1\))1411 3357 y Fs(\()p Fo(F)s(;)15 b(g)s Fs(\))1628 3284 y Fi(\001)1672 3357 y Fs(\()p Fo(V)20 b Fs(\))1815 3371 y Fk(x)1885 3357 y Fs(=)2029 3244 y Fn(2)1981 3271 y Fi(X)1984 3465 y Fk(r)r Fn(=1)2137 3296 y Fo(@)5 b(g)s Fs(\()p Fo(j)2313 3263 y Fn(2)2308 3318 y Fk(x)2355 3296 y Fo(F)13 b Fs(\))p 2137 3336 324 4 v 2160 3420 a Fo(@)5 b(y)2261 3376 y Fk(\015)2258 3447 y(i)2282 3456 y Fb(1)2317 3447 y Fe(\001\001\001)n Fk(i)2399 3455 y Fg(r)2613 3296 y Fo(@)2666 3263 y Fk(r)2704 3296 y Fo(V)2777 3263 y Fk(\015)p 2481 3336 473 4 v 2481 3420 a Fo(@)g(x)2586 3393 y Fk(i)2610 3402 y Fb(1)2664 3420 y Fm(\001)15 b(\001)g(\001)h Fo(@)5 b(x)2890 3393 y Fk(i)2914 3401 y Fg(r)2963 3357 y Fo(:)299 3636 y Fs(By)31 b(the)f(Leibniz)f(rule,)505 3832 y Fi(X)482 4034 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)675 3919 y Fo(D)753 3881 y Fk(\013)802 3845 y Fi(\000)844 3919 y Fo(C)916 3881 y Fn(\(1\))1010 3919 y Fs(\()p Fo(F)s(;)15 b(g)s Fs(\))1227 3845 y Fi(\001)1270 3919 y Fs(\()p Fo(V)21 b Fs(\))1414 3933 y Fk(x)1483 3919 y Fs(=)1627 3805 y Fn(2)1579 3832 y Fi(X)1583 4027 y Fk(r)r Fn(=1)1749 3832 y Fi(X)1726 4034 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)1925 3832 y Fi(X)1919 4030 y Fk(\014)s Fe(\024)p Fk(\013)2077 3919 y Fo(c)2116 3934 y Fk(\013\014)2209 3919 y Fo(D)2287 3881 y Fk(\013)p Fe(\000)p Fk(\014)2434 3763 y Fi( )2516 3857 y Fo(@)5 b(g)s Fs(\()p Fo(j)2692 3824 y Fn(2)2687 3880 y Fk(x)2733 3857 y Fo(F)13 b Fs(\))p 2516 3898 324 4 v 2539 3981 a Fo(@)5 b(y)2640 3937 y Fk(\015)2637 4009 y(i)2661 4018 y Fb(1)2695 4009 y Fe(\001\001\001)o Fk(i)2778 4017 y Fg(r)2850 3763 y Fi(!)2922 3919 y Fo(D)3000 3881 y Fk(\013)3049 3763 y Fi( )3263 3857 y Fo(@)3316 3824 y Fk(r)3354 3857 y Fo(V)3427 3824 y Fk(\015)p 3131 3898 473 4 v 3131 3981 a Fo(@)g(x)3236 3955 y Fk(i)3260 3964 y Fb(1)3314 3981 y Fm(\001)15 b(\001)g(\001)h Fo(@)5 b(x)3540 3955 y Fk(i)3564 3963 y Fg(r)3613 3763 y Fi(!)3685 3919 y Fo(;)299 4221 y Fs(where)28 b Fo(\014)i Fm(\024)25 b Fo(\013)k Fs(means)f Fo(b)1139 4235 y Fk(i)1193 4221 y Fm(\024)d Fo(a)1337 4235 y Fk(i)1393 4221 y Fs(for)j(1)e Fm(\024)f Fo(i)g Fm(\024)g Fo(m)p Fs(,)k(and)f(the)g(n)m(um)m(b)s(ers)f Fo(c)2716 4236 y Fk(\013\014)2837 4221 y Fs(are)i(p)s(ositiv)m(e)e(in)m (tegers)i(whose)299 4334 y(exact)40 b(v)-5 b(alue)38 b(is)g(irrelev)-5 b(an)m(t)38 b(to)h(our)g(presen)m(t)f(needs.)66 b(W)-8 b(e)40 b(compute)f(the)g(\()p Fo(\013)26 b Fm(\000)g Fo(\014)5 b Fs(\)-th)39 b(deriv)-5 b(ativ)m(e)38 b(of)299 4447 y Fo(@)5 b(g)s Fs(\()p Fo(j)475 4414 y Fn(2)470 4470 y Fk(x)516 4447 y Fo(F)13 b Fs(\))p Fo(=@)5 b(y)768 4403 y Fk(\015)765 4474 y(i)789 4483 y Fb(1)825 4474 y Fe(\001\001\001)n Fk(i)907 4482 y Fg(r)977 4447 y Fs(b)m(y)30 b(using)f(the)h(c)m(hain)g(rule,)g(and)f(then)h(writing)f(the)h(result) g(in)f(the)h(form)712 4677 y Fi(X)688 4879 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)881 4764 y Fo(D)959 4726 y Fk(\013)1009 4690 y Fi(\000)1050 4764 y Fo(C)1122 4726 y Fn(\(1\))1216 4764 y Fs(\()p Fo(F)s(;)15 b(g)s Fs(\))1433 4690 y Fi(\001)1477 4764 y Fs(\()p Fo(V)20 b Fs(\))1620 4778 y Fk(x)1690 4764 y Fs(=)1786 4641 y Fn([)p Fk(k)r(=)p Fn(2])p Fe(\000)p Fn(1)1839 4677 y Fi(X)1848 4873 y Fk(i)p Fn(=0)2196 4677 y Fi(X)2039 4879 y Fe(j)p Fk(\013)p Fn(+)p Fk(\014)s Fe(j\024)p Fk(k)r Fn(+2)p Fe(\000)p Fk(i)2499 4764 y Fo(g)2542 4779 y Fk(i;\013\014)2679 4764 y Fs(\()p Fo(j)2756 4726 y Fk(i)p Fn(+2)2751 4786 y Fk(x)2875 4764 y Fo(F)13 b Fs(\)\()p Fo(D)3094 4726 y Fk(\013)3144 4764 y Fo(F)s(;)i(D)3323 4726 y Fk(\014)3370 4764 y Fo(V)21 b Fs(\))p Fo(;)299 5071 y Fs(where)k(the)h Fo(g)752 5086 y Fk(i)p Fn(+2)p Fk(;\013\014)1004 5071 y Fs(are)g Fo(H)7 b(om)p Fs(\()p Fl(R)1471 5038 y Fk(n)1529 5071 y Fm(\002)k Fl(R)1689 5038 y Fk(n)1736 5071 y Fo(;)k Fl(R)1854 5038 y Fk(q)1893 5071 y Fs(\)-v)-5 b(alued)25 b(functions)f(de\014ned)g(on)i(W.)g(These) f(are)h(at)g(least)299 5184 y Fo(C)371 5151 y Fn(1)440 5184 y Fs(since)k(they)g(are)h(giv)m(en)g(in)e(terms)h(of)g(the)h (\014rst)f Fo(k)23 b Fs(+)d(1)31 b(deriv)-5 b(ativ)m(es)29 b(of)i Fo(g)s Fs(,)g(whic)m(h)e(is)h Fo(C)3383 5151 y Fk(k)r Fn(+)p Fk(l)q Fn(+1)3592 5184 y Fs(.)299 5396 y Fp(Step)j(1)p eop %%Page: 14 22 14 21 bop 0 100 a Fs(14)136 490 y(Let)29 b Fo(F)39 b Fm(2)25 b(E)536 452 y Fk(k)r Fn(+2)528 518 y(\012)697 490 y Fs(and)j Fo(g)h Fm(2)c Fo(C)1102 457 y Fk(k)r Fn(+)p Fk(l)q Fn(+1)1311 490 y Fs(\()p Fo(J)1405 457 y Fn(2)1396 517 y(\012)1452 490 y Fo(;)15 b Fl(R)1571 457 y Fk(q)1609 490 y Fs(\).)40 b(Consider)27 b(a)i(sequence)g Fm(f)p Fo(F)2639 505 y Fk(l)2666 490 y Fm(g)c(\032)g(E)2880 504 y Fn(\012)2964 490 y Fs(con)m(v)m(erging)k(to)h Fo(F)0 603 y Fs(in)e(the)h(\000)317 570 y Fk(k)r Fn(+2)450 603 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)697 570 y Fk(n)744 603 y Fs(\))30 b(top)s(ology)-8 b(.)41 b(W)-8 b(e)30 b(sho)m(w)f(that)h (for)f(an)m(y)g(set)h(of)g(v)m(ectors)g Fo(V)2697 617 y Fn(1)2737 603 y Fo(;)15 b(:::;)g(V)2945 617 y Fk(j)3008 603 y Fm(2)25 b Fo(C)3166 570 y Fe(1)3240 603 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3487 570 y Fk(n)3534 603 y Fs(\),)0 716 y(w)m(e)45 b(ha)m(v)m(e)371 642 y Fi(\000)413 716 y Fo(C)485 683 y Fn(\()p Fk(j)t Fn(\))576 716 y Fs(\()p Fo(F)669 731 y Fk(l)696 716 y Fo(;)15 b(g)s Fs(\))817 642 y Fi(\001)860 716 y Fs(\()p Fo(V)948 730 y Fn(1)987 716 y Fo(;)g(:::;)g(V)1195 730 y Fk(j)1234 716 y Fs(\))48 b Fm(2)g Fs(\000)1483 683 y Fk(k)1526 716 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1773 683 y Fk(q)1811 716 y Fs(\).)83 b(Moreo)m(v)m(er,)50 b(w)m(e)45 b(also)f(sho)m(w)g(that)h Fo(C)3262 683 y Fn(\()p Fk(j)t Fn(\))3353 716 y Fs(\()p Fo(F)3446 731 y Fk(l)3472 716 y Fo(;)15 b(g)s Fs(\))0 829 y(de\014nes)32 b(a)h(b)s(ounded)d(op)s(erator)j(in)f Fl(B)p Fs(\(\000)1396 796 y Fk(k)r Fn(+2)1529 829 y Fs(\()p Fo(M)5 b(;)15 b(R)1767 796 y Fk(n)1815 829 y Fs(\))22 b Fm(\002)f(\001)15 b(\001)g(\001)23 b(\002)e Fs(\000)2241 796 y Fk(k)r Fn(+2)2374 829 y Fs(\()p Fo(M)5 b(;)15 b(R)2612 796 y Fk(n)2659 829 y Fs(\))p 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Fm(\024)g Fs([)p Fo(k)s(=)p Fs(2])26 b Fm(\000)e Fs(1)h(+)g(2,)39 b(since)d(the)h(sum)f(o)m(v)m(er)i Fo(i)f Fs(runs)e(up)h(to)i([)p Fo(k)s(=)p Fs(2])26 b Fm(\000)e Fs(1.)0 1845 y(Because)43 b Fo(k)48 b(>)c Fs(2,)h(w)m(e)e(ha) m(v)m(e)g(that)f([)p Fo(k)s(=)p Fs(2])30 b Fm(\000)d Fs(1)i(+)e(2)45 b Fm(\024)f Fo(k)31 b Fs(+)d(2)g Fm(\000)g Fs(1)g Fm(\000)g Fs([)2510 1809 y Fk(m)p Fe(\000)p Fn(1)p 2510 1824 153 4 v 2568 1876 a(2)2672 1845 y Fs(])45 b(=)f Fo(k)31 b Fs(+)d(2)g Fm(\000)g Fo(k)3253 1859 y Fn(0)3292 1845 y Fs(,)45 b(where)0 1958 y Fo(k)47 1972 y Fn(0)134 1958 y Fs(=)i(1)30 b(+)f([)462 1922 y Fk(m)p Fe(\000)p Fn(1)p 462 1937 V 521 1989 a(2)624 1958 y Fs(].)81 b(By)44 b(Sob)s(olev's)f(Lemma,)48 b(the)43 b(sequence)i Fm(f)p Fo(F)2354 1973 y Fk(l)2381 1958 y Fm(g)f Fs(is)e(a)i(con)m(v)m(erging)h (sequence)f(in)0 2079 y Fo(C)72 2046 y Fk(k)r Fn(+2)p Fe(\000)p Fk(k)293 2055 y Fb(0)331 2079 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)578 2046 y Fk(n)625 2079 y Fs(\).)53 b(Consequen)m(tly)-8 b(,)35 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Fn(+2)1713 3434 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1960 3401 y Fk(n)2007 3434 y Fs(\))9 b Fm(\002)g(\001)15 b(\001)g(\001)10 b(\002)f Fs(\000)2383 3401 y Fk(k)r Fn(+2)2515 3434 y Fs(\()p Fo(M)c(;)15 b Fl(R)2763 3401 y Fk(n)2809 3434 y Fs(\))p Fo(;)g Fs(\000)2941 3401 y Fk(k)2985 3434 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3232 3401 y Fk(q)3270 3434 y Fs(\)\))25 b(for)g(all)0 3547 y Fo(l)30 b Fm(2)e Fl(N)p Fs(,)33 b(and)e(that)i Fm(f)p Fo(C)780 3514 y Fn(\()p Fk(j)t Fn(\))871 3547 y Fs(\()p Fo(F)964 3562 y Fk(l)991 3547 y Fo(;)15 b(g)s Fs(\))p Fm(g)34 b Fs(is)d(a)i(Cauc)m(h)m(y)f(sequence.)47 b(W)-8 b(e)33 b(start)g(b)m(y)f(estimating)g(the)g(di\013erence)0 3587 y Fi(\000)42 3660 y Fo(C)114 3627 y Fn(\()p Fk(j)t Fn(\))205 3660 y Fs(\()p Fo(F)298 3675 y Fk(l)324 3660 y Fo(;)15 b(g)s Fs(\))22 b Fm(\000)e Fo(C)630 3627 y Fn(\()p Fk(j)t Fn(\))721 3660 y Fs(\()p Fo(F)814 3674 y Fk(m)881 3660 y Fo(;)15 b(g)s Fs(\))1002 3587 y Fi(\001)1045 3660 y Fs(\()p Fo(V)1133 3674 y Fn(1)1173 3660 y Fo(;)g(:::;)32 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Fk(n)1746 5284 y Fo(;)g Fl(R)1865 5251 y Fk(q)1903 5284 y Fs(\))22 b(denotes)g(the)h(di\013eren)m(tial)d (of)i Fo(g)3013 5298 y Fk(i;\013)3102 5307 y Fb(0)3137 5298 y Fe(\001\001\001)o Fk(\013)3241 5308 y Fg(j)3300 5284 y Fs(and)g(the)0 5396 y(map)f Fo(\016)236 5363 y Fk(i)p Fn(+2)355 5396 y Fo(F)413 5411 y Fk(l)465 5396 y Fs(:)k Fo(U)35 b Fm(!)25 b Fo(T)13 b(W)34 b Fs(is)20 b(de\014ned)f(b)m(y)i(\(3)p Fo(:)p Fs(1\).)40 b(By)21 b(h)m(yp)s(othesis,)h(the)f(sequence)g Fm(f)p Fo(F)2882 5411 y Fk(l)2909 5396 y Fm(g)26 b(\032)f Fs(\000)3133 5363 y Fk(k)r Fn(+2)3265 5396 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3513 5363 y Fk(n)3559 5396 y Fs(\))p eop %%Page: 15 23 15 22 bop 3803 100 a Fs(15)299 490 y(is)25 b(Cauc)m(h)m(y)h(and)g(b)m (y)f(Sob)s(olev's)h(Lemma,)h(it)e(is)g(also)h(Cauc)m(h)m(y)h(in)d Fo(C)2589 457 y Fk(k)r Fn(+2)p Fe(\000)p Fk(k)2810 466 y Fb(0)2874 490 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3121 457 y Fk(n)3168 490 y Fs(\).)39 b(Therefore,)27 b(since)299 603 y Fo(i)21 b Fs(+)g(2)29 b Fm(\024)e Fo(k)e Fs(+)c(2)g Fm(\000)g Fo(k)984 617 y Fn(0)1024 603 y Fs(,)33 b(the)f(con)m(v)m(ex)h (h)m(ull)d(of)i(the)h(set)f Fm(f)p Fo(j)2214 570 y Fk(i)p Fn(+2)2209 625 y Fk(x)2334 603 y Fo(F)2392 618 y Fk(l)2446 603 y Fm(2)27 b Fo(J)2593 565 y Fk(i)p Fn(+2)2584 631 y(\012)2740 603 y Fs(:)h Fo(l)i Fm(2)d Fl(N)p Fo(;)k(x)c Fm(2)h Fo(M)10 b Fm(g)32 b Fs(is)f(con)m(tained)299 716 y(in)g(a)i(compact)g(subset)f Fo(K)j Fm(\032)29 b Fo(J)1402 677 y Fk(i)p Fn(+2)1393 744 y(\012)1520 716 y Fs(,)k(and)f(then)g(the)g (maps)g Fo(g)2407 730 y Fk(i;\013)2496 739 y Fb(0)2531 730 y Fe(\001\001\001)o Fk(\013)2635 740 y Fg(j)2672 716 y Fo(;)f(dg)2818 730 y Fk(i;\013)2907 739 y Fb(0)2942 730 y Fe(\001\001\001)o Fk(\013)3046 740 y Fg(j)3115 716 y Fs(are)i(b)s(ounded)d(on)i Fo(W)13 b Fs(.)299 829 y(W)-8 b(e)38 b(no)m(w)g(use)f(\(3)p Fo(:)p Fs(4\))i(and)e(\(3)p Fo(:)p Fs(5\))i(to)f(obtain)f(a)h(global)f(estimate)h(in)e(the)i(\000) 2926 796 y Fk(k)2968 829 y Fs(-norm)f(of)h(the)f(di\013erence)299 868 y Fi(\000)341 941 y Fo(C)413 908 y Fn(\()p Fk(j)t Fn(\))503 941 y Fs(\()p Fo(F)596 956 y Fk(l)623 941 y Fo(;)15 b(g)s Fs(\))22 b Fm(\000)e Fo(C)929 908 y Fn(\()p Fk(j)t Fn(\))1020 941 y Fs(\()p Fo(F)1113 955 y Fk(m)1180 941 y Fo(;)15 b(g)s Fs(\))1301 868 y Fi(\001)1344 941 y Fs(\()p Fo(V)1432 955 y Fn(1)1472 941 y Fo(;)g(:::;)g(V)1680 955 y Fk(j)1718 941 y Fs(\))31 b(using)e(a)h(partition)f(of)i(unit)m(y) -8 b(.)40 b(This)29 b(giv)m(es)603 1160 y Fm(jj)653 1087 y Fi(\000)695 1160 y Fo(C)767 1123 y Fn(\()p Fk(j)t Fn(\))858 1160 y Fs(\()p Fo(F)951 1175 y Fk(l)978 1160 y Fo(;)15 b(g)s Fs(\))22 b Fm(\000)e Fo(C)1284 1123 y Fn(\()p Fk(j)t Fn(\))1374 1160 y Fs(\()p Fo(F)1467 1174 y Fk(m)1535 1160 y Fo(;)15 b(g)s Fs(\))1656 1087 y Fi(\001)1699 1160 y Fs(\()p Fo(V)1787 1174 y Fn(1)1827 1160 y Fo(;)g(:::;)g(V)2035 1174 y Fk(j)2073 1160 y Fs(\))p Fm(jj)2158 1175 y Fk(k)2227 1160 y Fm(\024)25 b Fo(c)2362 1174 y Fn(1)2401 1160 y Fm(jj)p Fo(F)2509 1175 y Fk(l)2556 1160 y Fm(\000)20 b Fo(F)2705 1174 y Fk(m)2772 1160 y Fm(jj)2822 1175 y Fk(k)r Fn(+2)2956 1160 y Fm(jj)p Fo(V)3059 1174 y Fn(1)3099 1160 y Fm(jj)3149 1175 y Fk(k)r Fn(+2)3297 1160 y Fm(\001)15 b(\001)g(\001)h(jj)p Fo(V)3521 1174 y Fk(j)3558 1160 y Fm(jj)3608 1175 y Fk(k)r Fn(+2)1724 1298 y Fs(+)p Fo(c)1834 1312 y Fn(2)1874 1298 y Fm(jj)p Fo(F)1982 1313 y Fk(l)2029 1298 y Fm(\000)k Fo(F)2178 1312 y Fk(m)2245 1298 y Fm(jj)2295 1323 y Fk(C)2350 1301 y Fg(k)q Fb(+2)p Fa(\000)p Fg(k)2542 1316 y Fb(0)2585 1298 y Fm(jj)p Fo(F)2693 1312 y Fk(m)2761 1298 y Fm(jj)2811 1313 y Fk(k)r Fn(+2)2944 1298 y Fm(jj)p Fo(V)3047 1312 y Fn(1)3087 1298 y Fm(jj)3137 1313 y Fk(k)r Fn(+2)3286 1298 y Fm(\001)15 b(\001)g(\001)h(jj)p Fo(V)3510 1312 y Fk(j)3547 1298 y Fm(jj)3597 1313 y Fk(k)r Fn(+2)3730 1298 y Fo(;)299 1502 y Fs(for)33 b(p)s(ositiv)m(e)f Fo(c)819 1516 y Fn(1)858 1502 y Fo(;)15 b(c)937 1516 y Fn(2)977 1502 y Fs(.)49 b(Since)32 b Fm(fjj)p Fo(F)1444 1516 y Fk(m)1512 1502 y Fm(jj)1562 1517 y Fk(k)r Fn(+2)1695 1502 y Fm(g)e(\032)f Fl(R)34 b Fs(is)e(a)h(b)s(ounded)e(sequence)i(of)g (real)g(n)m(um)m(b)s(ers,)f(and)g(the)299 1615 y(inclusion)f(\000)739 1582 y Fk(k)r Fn(+2)872 1615 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1119 1582 y Fk(n)1166 1615 y Fs(\))32 b Fo(,)-15 b Fm(!)31 b Fo(C)1437 1582 y Fk(k)r Fn(+2)p Fe(\000)p Fk(k)1658 1591 y Fb(0)1695 1615 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1943 1582 y Fk(n)1990 1615 y Fs(\))34 b(is)f(con)m(tin)m(uous,)i (the)f(last)g(term)g(ab)s(o)m(v)m(e)i(is)d(b)s(ounded)299 1728 y(b)m(y)f(a)h(p)s(ositiv)m(e)f(m)m(ultiple)e(of)j Fm(jj)p Fo(F)1416 1743 y Fk(l)1465 1728 y Fm(\000)21 b Fo(F)1615 1742 y Fk(m)1682 1728 y Fm(jj)1732 1743 y Fk(k)r Fn(+2)1865 1728 y Fm(jj)p Fo(V)1968 1742 y Fn(1)2008 1728 y Fm(jj)2058 1743 y Fk(k)r Fn(+2)2207 1728 y Fm(\001)15 b(\001)g(\001)h(jj)p Fo(V)2431 1742 y Fk(j)2468 1728 y Fm(jj)2518 1743 y Fk(k)r Fn(+2)2652 1728 y Fs(.)47 b(This)31 b(sho)m(ws)h(that)h Fo(C)3468 1695 y Fn(\()p Fk(j)t Fn(\))3559 1728 y Fs(\()p Fo(F)3652 1743 y Fk(l)3679 1728 y Fo(;)15 b(g)s Fs(\))23 b Fm(\000)299 1841 y Fo(C)371 1808 y Fn(\()p Fk(j)t Fn(\))462 1841 y Fs(\()p Fo(F)555 1855 y Fk(m)622 1841 y Fo(;)15 b(g)s Fs(\))23 b(is)e(a)g(b)s(ounded)f (op)s(erator)i(in)e Fl(B)p Fs(\(\000)1899 1808 y Fk(k)r Fn(+2)2032 1841 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2279 1808 y Fk(n)2326 1841 y Fs(\))s Fm(\002)s(\001)g(\001)g(\001)s(\002)s Fs(\000)2677 1808 y Fk(k)r Fn(+2)2809 1841 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3056 1808 y Fk(n)3103 1841 y Fs(\))p Fo(;)g Fs(\000)3235 1808 y Fk(k)3278 1841 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3526 1808 y Fk(q)3564 1841 y Fs(\)\))22 b(whose)299 1954 y(op)s(erator)38 b(norm)f(is)f(b)s(ounded)g(b)m(y)h(a) h(p)s(ositiv)m(e)f(m)m(ultiple)e(of)j Fm(jj)p Fo(F)2537 1969 y Fk(l)2589 1954 y Fm(\000)24 b Fo(F)2742 1968 y Fk(m)2809 1954 y Fm(jj)2859 1969 y Fk(k)r Fn(+2)2993 1954 y Fs(.)62 b(This)36 b(sho)m(ws)h(that)h(the)299 2067 y(sequence)511 2271 y Fm(f)p Fo(C)628 2234 y Fn(\()p Fk(j)t Fn(\))719 2271 y Fs(\()p Fo(F)812 2286 y Fk(l)839 2271 y Fo(;)15 b(g)s Fs(\))21 b Fm(\000)f Fo(C)1144 2234 y Fn(\()p Fk(j)t Fn(\))1235 2271 y Fs(\()p Fo(F)1328 2285 y Fk(m)1395 2271 y Fo(;)15 b(g)s Fs(\))p Fm(g)28 b(\032)d Fl(B)p Fs(\(\000)1851 2234 y Fk(k)r Fn(+2)1984 2271 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2231 2234 y Fk(n)2278 2271 y Fs(\))21 b Fm(\002)f(\001)15 b(\001)g(\001)21 b(\002)f Fs(\000)2699 2234 y Fk(k)r Fn(+2)2831 2271 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3079 2234 y Fk(n)3126 2271 y Fs(\))p Fo(;)g Fs(\000)3258 2234 y Fk(k)3301 2271 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3548 2234 y Fk(q)3586 2271 y Fs(\)\))p Fo(;)299 2475 y Fs(is)29 b(Cauc)m(h)m(y)-8 b(.)42 b(This)28 b(concludes)i(the)h(pro)s(of)e(of)i(Step)f(1.)p eop %%Page: 16 24 16 23 bop 0 100 a Fs(16)0 490 y Fp(Step)33 b(2)136 681 y Fs(Fix)d Fo(g)f Fm(2)c Fo(C)529 648 y Fk(k)r Fn(+)p Fk(l)q Fn(+1)738 681 y Fs(\()p Fo(J)832 648 y Fn(2)823 708 y(\012)879 681 y Fo(;)15 b Fl(R)998 648 y Fk(q)1036 681 y Fs(\).)41 b(F)-8 b(or)31 b Fo(j)g Fs(=)25 b(1,)31 b(w)m(e)g(sho)m(w)f(that)h(the)f(limit)795 934 y(lim)793 992 y Fk(t)p Fe(!)p Fn(0)939 775 y Fi(\014)939 829 y(\014)939 884 y(\014)939 938 y(\014)939 993 y(\014)969 775 y(\014)969 829 y(\014)969 884 y(\014)969 938 y(\014)969 993 y(\014)1010 872 y Fo(C)7 b Fs(\()p Fo(F)33 b Fs(+)20 b Fo(tV)5 b(;)15 b(g)s Fs(\))21 b Fm(\000)f Fo(C)7 b Fs(\()p Fo(F)s(;)15 b(g)s Fs(\))p 1010 913 904 4 v 1445 996 a Fo(t)1943 934 y Fm(\000)20 b Fo(C)2106 896 y Fn(\(1\))2200 934 y Fs(\()p Fo(F)s(;)15 b(g)s Fs(\))p Fo(V)2492 775 y Fi(\014)2492 829 y(\014)2492 884 y(\014)2492 938 y(\014)2492 993 y(\014)2522 775 y(\014)2522 829 y(\014)2522 884 y(\014)2522 938 y(\014)2522 993 y(\014)2553 1052 y Fk(k)2621 934 y Fs(=)24 b(0)p Fo(;)-2786 b Fs(\(3.6\))0 1187 y(for)30 b(all)f Fo(F)39 b Fm(2)25 b(E)496 1201 y Fn(\012)551 1187 y Fo(;)30 b(V)46 b Fm(2)25 b Fo(C)863 1154 y Fe(1)937 1187 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1184 1154 y Fk(n)1231 1187 y Fs(\).)41 b(Since)29 b(the)i(map)845 1359 y Fo(C)917 1321 y Fn(\(1\))1011 1359 y Fs(\()p Fm(\001)p Fo(;)15 b(g)s Fs(\))27 b(:)f Fm(E)1326 1320 y Fk(k)r Fn(+2)1318 1387 y(\012)1484 1359 y Fm(!)f Fl(B)p Fs(\(\000)1766 1321 y Fk(k)r Fn(+2)1899 1359 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2146 1321 y Fk(n)2193 1359 y Fs(\))p Fo(;)g Fs(\000)2325 1321 y Fk(k)2369 1359 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2616 1321 y Fk(q)2654 1359 y Fs(\)\))p Fo(;)0 1530 y Fs(is)29 b(con)m(tin)m(uous,)i(w)m(e)g (conclude)e(that)i Fo(C)1348 1497 y Fn(\(1\))1442 1530 y Fs(\()p Fm(\001)p Fo(;)15 b(g)s Fs(\))32 b(is)e(the)g(deriv)-5 b(ativ)m(e)30 b(of)h Fo(C)7 b Fs(\()p Fm(\001)p Fo(;)15 b(g)s Fs(\))32 b(\([12)q(],)f(p.37\).)136 1643 y(T)-8 b(o)23 b(pro)m(v)m(e)g(\(3)p Fo(:)p Fs(6\),)j(w)m(e)d(consider)e(a)i (co)m(v)m(er)h(of)e Fo(M)33 b Fs(b)m(y)22 b(lo)s(cal)g(c)m(harts.)38 b(In)22 b(eac)m(h)h(of)g(these)g(c)m(harts)f Fo(U)36 b Fm(\032)25 b Fl(R)3503 1610 y Fk(m)3570 1643 y Fs(,)0 1756 y(w)m(e)31 b(ha)m(v)m(e)g(the)g(p)s(oin)m(t)m(wise)e(limit)593 2000 y(lim)590 2058 y Fk(t)p Fe(!)p Fn(0)760 1914 y Fi(X)737 2115 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)930 2000 y Fo(D)1008 1963 y Fk(\013)1057 1845 y Fi( )1139 1939 y Fo(g)s Fs(\()p Fo(j)1262 1906 y Fn(2)1257 1961 y Fk(x)1303 1939 y Fs(\()p Fo(F)34 b Fs(+)20 b Fo(tV)g Fs(\)\))h Fm(\000)f Fo(g)s Fs(\()p Fo(j)1932 1894 y Fk(p)1927 1950 y(x)1973 1939 y Fo(F)13 b Fs(\))p 1139 1979 941 4 v 1593 2062 a Fo(t)2110 2000 y Fm(\000)20 b Fo(dg)s Fs(\()p Fo(j)2371 1963 y Fn(2)2366 2023 y Fk(x)2412 2000 y Fo(F)13 b Fs(\))21 b Fm(\001)f Fo(\016)2627 1963 y Fn(2)2667 2000 y Fo(V)2740 1845 y Fi(!)2838 2000 y Fs(=)25 b(0)p Fo(:)-3004 b Fs(\(3.7\))0 2285 y(W)-8 b(e)25 b(can)g(write)e(the)h(quotien)m(t)1040 2212 y Fi(\000)1081 2285 y Fo(g)s Fs(\()p Fo(j)1204 2252 y Fn(2)1199 2308 y Fk(x)1246 2285 y Fs(\()p Fo(F)d Fs(+)8 b Fo(tV)19 b Fs(\)\))8 b Fm(\000)g Fo(g)s Fs(\()p Fo(j)1824 2252 y Fn(2)1819 2308 y Fk(x)1865 2285 y Fo(F)13 b Fs(\))1971 2212 y Fi(\001)2013 2285 y Fo(=t)25 b Fs(b)m(y)f(applying)e(the)i(Mean) h(V)-8 b(alue)24 b(Theorem)0 2398 y(to)31 b(the)g(function)e Fo(\036)p Fs(\()p Fo(t)p Fs(\))d(=)f Fo(g)s Fs(\()p Fo(j)1026 2365 y Fn(2)1021 2421 y Fk(x)1067 2398 y Fs(\()p Fo(F)33 b Fs(+)20 b Fo(tV)g Fs(\)\).)42 b(W)-8 b(e)31 b(get)821 2564 y Fo(g)s Fs(\()p Fo(j)944 2531 y Fn(2)939 2586 y Fk(x)985 2564 y Fs(\()p Fo(F)i Fs(+)20 b Fo(tV)g Fs(\)\))h Fm(\000)f Fo(g)s Fs(\()p Fo(j)1613 2531 y Fn(2)1608 2586 y Fk(x)1654 2564 y Fo(F)13 b Fs(\))p 821 2604 940 4 v 1274 2688 a Fo(t)1796 2625 y Fs(=)25 b Fo(dg)s Fs(\()p Fo(j)2062 2588 y Fn(2)2057 2648 y Fk(x)2103 2625 y Fs(\()p Fo(F)34 b Fs(+)20 b Fo(t)2354 2639 y Fn(1)2393 2625 y Fo(V)g Fs(\)\))h Fm(\001)f Fo(\016)2645 2588 y Fn(2)2686 2625 y Fo(V)5 b(;)-2769 b Fs(\(3.8\))0 2823 y(for)37 b(some)h Fm(j)p Fo(t)439 2837 y Fn(1)479 2823 y Fm(j)f(\024)g(j)p Fo(t)p Fm(j)p Fs(.)62 b(W)-8 b(e)39 b(apply)d(the)i(Leb)s(esgue)f (Dominated)h(Con)m(v)m(ergence)h(Theorem)e(to)i(the)e(se-)0 2936 y(quence)357 3024 y Fi(\()468 3093 y(X)445 3295 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)638 3180 y Fo(D)716 3142 y Fk(\013)765 3024 y Fi( )847 3118 y Fo(g)s Fs(\()p Fo(j)970 3085 y Fn(2)965 3141 y Fk(x)1011 3118 y Fs(\()p Fo(F)d Fs(+)20 b Fo(tV)g Fs(\)\))h Fm(\000)f Fo(g)s Fs(\()p Fo(j)1640 3085 y Fn(2)1635 3141 y Fk(x)1681 3118 y Fo(F)13 b Fs(\))p 847 3159 V 1301 3242 a Fo(t)1817 3180 y Fm(\000)20 b Fo(dg)s Fs(\()p Fo(j)2078 3142 y Fn(2)2073 3202 y Fk(x)2120 3180 y Fo(F)13 b Fs(\))20 b Fm(\001)h Fo(\016)2335 3142 y Fn(2)2375 3180 y Fo(V)2448 3024 y Fi(!\))2594 3297 y Fk(t)p Fe(2)p Fk(I)2731 3180 y Fm(\032)k Fs(\000)2884 3142 y Fn(0)2923 3180 y Fs(\()p Fo(U;)15 b Fl(R)3140 3142 y Fk(q)3178 3180 y Fs(\))p Fo(;)0 3463 y Fs(to)37 b(sho)m(w)f(that)h(the)f(limit)e(\(3)p Fo(:)p Fs(7\))k(holds)c(in)h (the)h(\000)1733 3430 y Fn(0)1773 3463 y Fs(-norm.)57 b(F)-8 b(or)37 b(this,)g(it)e(su\016ces)h(to)h(\014nd)e(a)h(function)0 3576 y Fo(G)26 b Fm(2)f Fs(\000)240 3543 y Fn(0)279 3576 y Fs(\()p Fo(U;)15 b Fl(R)q Fs(\))31 b(suc)m(h)f(that)1076 3666 y Fi(\014)1076 3721 y(\014)1076 3775 y(\014)1076 3830 y(\014)1076 3884 y(\014)1145 3739 y(X)1122 3940 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)1315 3825 y Fo(D)1393 3788 y Fk(\013)1442 3825 y Fo(dg)s Fs(\()p Fo(j)1612 3788 y Fn(2)1607 3848 y Fk(x)1654 3825 y Fs(\()p Fo(F)j Fs(+)20 b Fo(tV)g Fs(\)\))h Fm(\001)g Fo(\016)2157 3788 y Fn(2)2197 3825 y Fo(V)2270 3666 y Fi(\014)2270 3721 y(\014)2270 3775 y(\014)2270 3830 y(\014)2270 3884 y(\014)2326 3825 y Fm(\024)k Fo(G;)0 4101 y Fs(for)30 b(all)f Fo(t)c Fm(2)g Fo(I)7 b Fs(,)31 b(since)f(b)m(y)g(\(3)p Fo(:)p Fs(8\))i(w)m(e)f(will)d(ha)m(v)m(e)590 4191 y Fi(\014)590 4245 y(\014)590 4300 y(\014)590 4355 y(\014)590 4409 y(\014)659 4264 y(X)636 4465 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)829 4350 y Fo(D)907 4313 y Fk(\013)956 4195 y Fi( )1038 4289 y Fo(g)s Fs(\()p Fo(j)1161 4256 y Fn(2)1156 4311 y Fk(x)1202 4289 y Fs(\()p Fo(F)34 b Fs(+)20 b Fo(tV)g Fs(\)\))h Fm(\000)e Fo(g)s Fs(\()p Fo(j)1830 4256 y Fn(2)1825 4311 y Fk(x)1871 4289 y Fo(F)13 b Fs(\))p 1038 4329 V 1492 4412 a Fo(t)2008 4350 y Fm(\000)20 b Fo(dg)s Fs(\()p Fo(j)2269 4313 y Fn(2)2264 4373 y Fk(x)2310 4350 y Fo(F)13 b Fs(\))21 b Fm(\001)f Fo(\016)2525 4313 y Fn(2)2566 4350 y Fo(V)2639 4195 y Fi(!)2711 4191 y(\014)2711 4245 y(\014)2711 4300 y(\014)2711 4355 y(\014)2711 4409 y(\014)2766 4350 y Fm(\024)25 b Fs(2)p Fo(G:)0 4634 y Fs(T)-8 b(o)39 b(construct)f(suc)m(h)g(function)f Fo(G)p Fs(,)k(w)m(e)d(giv)m(e)h(a)g (uniform)d(estimate)j(on)f Fo(t)g Fs(of)g Fm(j)2760 4565 y Fi(P)2856 4660 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)3054 4634 y Fo(D)3132 4601 y Fk(\013)3181 4634 y Fo(dg)s Fs(\()p Fo(j)3351 4601 y Fn(2)3346 4656 y Fk(x)3392 4634 y Fs(\()p Fo(F)h Fs(+)0 4761 y Fo(tV)20 b Fs(\)\))h Fm(\001)f Fo(\016)285 4728 y Fn(2)326 4761 y Fo(V)g Fm(j)p Fs(.)41 b(By)31 b(form)m(ula)e(\(3)p Fo(:)p Fs(3\),)k(w)m(e)d(ha)m(v)m(e)787 4860 y Fi(X)764 5061 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)957 4946 y Fo(D)1035 4909 y Fk(\013)1084 4946 y Fo(dg)s Fs(\()p Fo(j)1254 4909 y Fn(2)1249 4969 y Fk(x)1295 4946 y Fs(\()p Fo(F)k Fs(+)20 b Fo(tV)g Fs(\)\))h Fm(\001)f Fo(\016)1798 4909 y Fn(2)1838 4946 y Fo(V)610 5301 y Fs(=)764 5179 y Fn([)p Fk(k)r(=)p Fn(2])p Fe(\000)p Fn(1)817 5215 y Fi(X)826 5410 y Fk(i)p Fn(=0)1174 5215 y Fi(X)1017 5416 y Fe(j)p Fk(\013)p Fn(+)p Fk(\014)s Fe(j\024)p Fk(k)r Fn(+2)p Fe(\000)p Fk(i)1477 5301 y Fo(g)1520 5316 y Fk(i;\013\014)1657 5301 y Fs(\()p Fo(j)1734 5264 y Fk(i)p Fn(+2)1729 5324 y Fk(x)1853 5301 y Fs(\()p Fo(F)34 b Fs(+)20 b Fo(tV)g Fs(\)\)\()p Fo(D)2360 5264 y Fk(\013)2410 5301 y Fs(\()p Fo(F)34 b Fs(+)19 b Fo(tV)i Fs(\))p Fo(;)15 b(D)2887 5264 y Fk(\014)2934 5301 y Fo(V)20 b Fs(\))p Fo(:)p eop %%Page: 17 25 17 24 bop 3803 100 a Fs(17)299 490 y(Since)36 b Fo(t)i Fs(v)-5 b(aries)37 b(within)e(a)j(small)e(in)m(terv)-5 b(al,)39 b(the)f(set)g Fm(f)p Fo(j)2258 457 y Fk(i)p Fn(+2)2253 512 y Fk(x)2377 490 y Fs(\()p Fo(F)h Fs(+)24 b Fo(tV)c Fs(\)\))38 b Fm(2)f Fo(J)2975 452 y Fk(i)p Fn(+2)2966 518 y(\012)3131 490 y Fs(:)h Fo(t)f Fm(2)g Fo(I)7 b(;)30 b(x)38 b Fm(2)f Fo(M)10 b Fm(g)38 b Fs(is)299 603 y(con)m(tained)33 b(in)f(a)h(compact)h(subset)e Fo(K)k Fm(\032)29 b Fo(J)1819 565 y Fk(i)p Fn(+2)1810 631 y(\012)1938 603 y Fs(,)34 b(and)e(then)g(the)h(functions)f Fo(g)2982 618 y Fk(i;\013\014)3151 603 y Fs(are)h(b)s(ounded)e(on)i(it.)299 716 y(W)-8 b(e)31 b(estimate)512 804 y Fi(\014)512 858 y(\014)512 913 y(\014)512 967 y(\014)512 1022 y(\014)580 876 y(X)557 1078 y Fe(j)p Fk(\013)p Fe(j\024)p Fk(k)750 963 y Fo(D)828 925 y Fk(\013)877 963 y Fo(dg)s Fs(\()p Fo(j)1047 925 y Fn(2)1042 985 y Fk(x)1089 963 y Fs(\()p Fo(F)i Fs(+)20 b Fo(tV)g Fs(\)\))h Fm(\001)g Fo(\016)1592 925 y Fn(2)1632 963 y Fo(V)1705 804 y Fi(\014)1705 858 y(\014)1705 913 y(\014)1705 967 y(\014)1705 1022 y(\014)1819 963 y Fm(\024)82 b Fo(c)2026 840 y Fn([)p Fk(k)r(=)p Fn(2])p Fe(\000)p Fn(1)2080 876 y Fi(X)2089 1072 y Fk(i)p Fn(=0)2437 876 y Fi(X)2280 1078 y Fe(j)p Fk(\013)p Fn(+)p Fk(\014)s Fe(j\024)p Fk(k)r Fn(+2)p Fe(\000)p Fk(i)2741 963 y Fm(j)p Fo(D)2844 925 y Fk(\013)2893 963 y Fs(\()p Fo(F)34 b Fs(+)20 b Fo(tV)g Fs(\))p Fm(jj)p Fo(D)3380 925 y Fk(\014)3427 963 y Fo(V)h Fm(j)1819 1318 y(\024)82 b Fo(c)2026 1195 y Fn([)p Fk(k)r(=)p Fn(2])p Fe(\000)p Fn(1)2080 1231 y Fi(X)2089 1427 y Fk(i)p Fn(=0)2437 1231 y Fi(X)2280 1433 y Fe(j)p Fk(\013)p Fn(+)p Fk(\014)s Fe(j\024)p Fk(k)r Fn(+2)p Fe(\000)p Fk(i)2725 1318 y Fs(\()p Fm(j)p Fo(D)2863 1280 y Fk(\013)2913 1318 y Fo(F)13 b Fm(j)21 b Fs(+)f Fm(j)p Fo(D)3224 1280 y Fk(\013)3273 1318 y Fo(V)g Fm(j)p Fs(\))p Fm(j)p Fo(D)3509 1280 y Fk(\014)3557 1318 y Fo(V)g Fm(j)p Fo(;)-3381 b Fs(\(3.9\))299 1587 y(for)30 b(some)h Fo(c)25 b(>)g Fs(0.)41 b(Set)1146 1838 y Fo(G)25 b Fs(=)g Fo(c)1392 1716 y Fn([)p Fk(k)r(=)p Fn(2])p Fe(\000)p Fn(1)1446 1751 y Fi(X)1455 1947 y Fk(i)p Fn(=0)1803 1751 y Fi(X)1646 1953 y Fe(j)p Fk(\013)p Fn(+)p Fk(\014)s Fe(j\024)p Fk(k)r Fn(+2)p Fe(\000)p Fk(i)2091 1838 y Fs(\()p Fm(j)p Fo(D)2229 1800 y Fk(\013)2279 1838 y Fo(F)13 b Fm(j)21 b Fs(+)f Fm(j)p Fo(D)2590 1800 y Fk(\013)2639 1838 y Fo(V)g Fm(j)p Fs(\))p Fm(j)p Fo(D)2875 1800 y Fk(\014)2923 1838 y Fo(V)g Fm(j)p Fo(:)299 2107 y Fs(A)26 b(computation)g(sho)m(ws)g(that)h(at)f(least)h(one)f(of)h (the)f(m)m(ultiindices)d Fo(\013;)31 b(\014)g Fs(is)25 b(less)g(or)i(equal)e(to)i Fo(k)15 b Fs(+)d(2)g Fm(\000)g Fo(k)3830 2121 y Fn(0)3868 2107 y Fs(.)299 2220 y(By)31 b(Sob)s(olev's)e(Lemma,)i(at)h(least)e(one)h(of)g Fo(D)1864 2187 y Fk(\013)1913 2220 y Fo(F)s(;)15 b(D)2092 2187 y Fk(\014)2140 2220 y Fo(V)50 b Fs(is)30 b(a)h(con)m(tin)m(uous)f (function.)40 b(This)29 b(sho)m(ws)h(that)299 2333 y Fo(G)g Fm(2)f Fs(\000)547 2300 y Fn(0)587 2333 y Fs(\()p Fo(U;)15 b Fl(R)q Fs(\).)49 b(Therefore,)33 b(the)h(limit)d(\(3)p Fo(:)p Fs(7\))j(holds)e(in)g(\000)2358 2300 y Fn(0)2397 2333 y Fs(\()p Fo(U;)15 b Fl(R)2613 2300 y Fk(q)2651 2333 y Fs(\))33 b(for)g(ev)m(ery)h(lo)s(cal)e(c)m(hart)i Fo(U)43 b Fs(of)33 b Fo(M)10 b Fs(.)299 2446 y(A)30 b(similar)e (argumen)m(t)j(sho)m(ws)f(that)h Fo(C)1633 2413 y Fn(\()p Fk(j)t Fn(\))1724 2446 y Fs(\()p Fm(\001)p Fo(;)15 b(g)s Fs(\))32 b(is)e(the)g Fo(j)5 b Fs(-deriv)-5 b(ativ)m(e)31 b(of)g Fo(C)7 b 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y Fk(d)299 693 y Fs(In)40 b(this)f(section)i(w)m(e)g(c)m(haracterize)i(the)d (critical)g(em)m(b)s(eddings)f(of)h(the)h(distance)g(functional)e Fo(E)3717 708 y Fk(d)3798 693 y Fs(b)m(y)299 806 y(computing)27 b(its)h(\014rst)g(v)-5 b(ariational)27 b(deriv)-5 b(ativ)m(e)28 b(at)h(an)f(\012-regular)g(em)m(b)s(edding)e Fo(F)13 b Fs(.)41 b(Consider)26 b(a)j(smo)s(oth)299 919 y(v)-5 b(ariation)29 b(of)i Fo(F)13 b Fs(,)31 b(that)g(is,)e(a)i(smo)s(oth)f (function)1612 1076 y(\010)25 b(:)g Fo(M)31 b Fm(\002)19 b Fs(\()p Fm(\000)p Fo(\017;)c Fs(+)p Fo(\017)p Fs(\))26 b Fm(!)f Fl(R)2508 1039 y Fk(n)2556 1076 y Fo(;)299 1234 y Fs(suc)m(h)39 b(that)i(\010\()p Fm(\001)p Fo(;)15 b Fs(0\))42 b(=)f Fo(F)13 b Fs(.)69 b(Di\013eren)m(tiating)40 b(\010)f(with)g(resp)s(ect)h(to)g Fo(t)g Fs(at)g Fo(t)h Fs(=)g(0)f(giv)m(es)g(a)h(v)m(ector)g(\014eld)299 1347 y Fo(V)62 b Fs(de\014ned)41 b(along)h Fo(F)13 b Fs(\()p Fo(M)d Fs(\).)77 b(W)-8 b(e)44 b(call)d Fo(V)63 b Fs(a)42 b Fp(variational)k(ve)-5 b(ctor)44 b(\014eld)p Fs(.)77 b(Let)42 b(\010)3137 1361 y Fk(t)3212 1347 y Fs(=)i(\010\()p Fm(\001)p Fo(;)15 b(t)p Fs(\))43 b(and)f(let)299 1460 y Fo(\025)p Fs(\()p Fm(\001)p Fo(;)15 b(t)p Fs(\))26 b(:)g(\012)f Fm(!)g Fo(M)40 b Fs(b)s(e)30 b(the)h(pro)5 b(jection)30 b(index)f(asso)s(ciated)h(with)g(\010)2596 1474 y Fk(t)2625 1460 y Fs(.)299 1609 y Fl(Theorem)k(3.3.1)46 b Fp(L)-5 b(et)37 b Fo(F)49 b Fp(b)-5 b(e)37 b(an)g Fs(\012)p Fp(-r)-5 b(e)g(gular)37 b(emb)-5 b(e)g(dding)38 b(and)f(let)g Fs(\010)f Fp(b)-5 b(e)36 b(a)h(smo)-5 b(oth)39 b(variation)f(of)f Fo(F)13 b Fp(.)299 1722 y(Then,)1409 1890 y Fo(dE)1523 1905 y Fk(d)1564 1890 y Fs([\010\()p Fm(\001)p Fo(;)i(t)p Fs(\)])p 1409 1931 441 4 v 1589 2014 a Fo(dt)1859 1792 y Fi(\014)1859 1847 y(\014)1859 1902 y(\014)1859 1956 y(\014)1859 2011 y(\014)1889 2069 y Fk(t)p Fn(=0)2034 1952 y Fs(=)2130 1828 y Fi(Z)2181 2034 y Fk(M)2260 1952 y Fm(h)p Fo(\034)2335 1966 y Fk(F)2394 1952 y Fo(;)g(V)20 b Fm(i)p Fo(dv)s(ol)2707 1967 y Fk(h)2753 1952 y Fo(;)-2479 b Fs(\(3.14\))299 2186 y Fp(wher)-5 b(e)34 b Fo(\034)596 2200 y Fk(F)679 2186 y Fs(:)26 b Fo(M)35 b Fm(!)25 b Ff(R)1047 2153 y Fk(n)1127 2186 y Fp(is)32 b(the)h(normal)i(ve)-5 b(ctor)33 b(\014eld)g(along)h Fo(F)46 b Fp(de\014ne)-5 b(d)33 b(by)1189 2380 y Fo(\034)1229 2394 y Fk(F)1287 2380 y Fs(\()p Fo(x)p Fs(\))26 b(=)f Fm(\000)1617 2256 y Fi(Z)1667 2463 y Fn(\012)1718 2477 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))1871 2380 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2731 2394 y Fk(v)2771 2380 y Fs(\))p Fo(v)f(dv)s(;)-2689 b Fs(\(3.15\))299 2606 y Fp(and)33 b Fo(V)53 b Fp(is)32 b(the)g(variational)j(ve)-5 b(ctor)33 b(\014eld)g(gener)-5 b(ate)g(d)33 b(by)g Fs(\010)p Fp(.)41 b(Mor)-5 b(e)g(over,)33 b(the)g(emb)-5 b(e)g(dding)33 b Fo(F)46 b Fp(is)32 b(critic)-5 b(al)299 2719 y(if)32 b(and)i(only)f(if)f(it)h(satis\014es)h(the)f(e)-5 b(quation)1951 2876 y Fo(\034)1991 2890 y Fk(F)2075 2876 y Fs(=)25 b(0)p Fo(:)299 3034 y Fp(Pr)-5 b(o)g(of:)47 b Fs(This)29 b(is)g(a)i(direct)f (calculation.)39 b(Set)31 b Fo(E)1935 3049 y Fk(d)1976 3034 y Fs(\()p Fo(t)p Fs(\))25 b(=)g Fo(E)2267 3049 y Fk(d)2308 3034 y Fs([\010\()p Fm(\001)p Fo(;)15 b(t)p Fs(\)].)42 b(Then,)445 3235 y Fo(E)517 3197 y Fe(0)512 3258 y Fk(d)552 3235 y Fs(\()p Fo(t)p Fs(\))84 b(=)919 3173 y Fo(d)p 902 3214 81 4 v 902 3297 a(dt)1002 3173 y Fs(1)p 1002 3214 46 4 v 1002 3297 a(2)1073 3111 y Fi(Z)1124 3317 y Fn(\012)1194 3235 y Fm(j)p Fl(y)20 b Fm(\000)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(t)p Fs(\))p Fo(;)g(t)p Fs(\))p Fm(j)1871 3197 y Fn(2)1912 3235 y Fo(p)p Fs(\()p Fl(y)p Fs(\))p Fo(d)p Fl(y)739 3509 y Fs(=)82 b Fm(\000)978 3386 y Fi(Z)1029 3592 y Fn(\012)1099 3354 y Fi(*)1172 3509 y Fl(y)21 b Fm(\000)f Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b(t)p Fs(\))p Fo(;)g(t)p Fs(\))p Fo(;)1851 3448 y(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(t)p Fs(\))p Fo(;)g(t)p Fs(\))p 1851 3488 516 4 v 2042 3572 a Fo(@)5 b(x)2147 3545 y Fk(i)2386 3448 y Fo(@)g(\025)2492 3415 y Fk(i)2520 3448 y Fs(\()p Fl(y)q Fo(;)15 b(t)p Fs(\))p 2386 3488 334 4 v 2509 3572 a Fo(@)5 b(t)2750 3509 y Fs(+)2850 3448 y Fo(@)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(t)p Fs(\))p Fo(;)g(t)p Fs(\))p 2850 3488 516 4 v 3065 3572 a Fo(@)5 b(t)3375 3354 y Fi(+)3449 3509 y Fo(p)p Fs(\()p Fl(y)p Fs(\))p Fo(d)p Fl(y)q Fo(:)299 3753 y Fs(Note)42 b(that)g(for)e Fl(y)j Fm(2)f Fs(\012,)i(the)d(v)m(ector)h Fo(@)5 b Fs(\010)1791 3767 y Fk(t)1821 3753 y Fo(=@)g(x)1971 3720 y Fk(i)2041 3753 y Fs(is)39 b(tangen)m(tial)j(to)f(\010)2767 3767 y Fk(t)2797 3753 y Fs(\()p Fo(M)10 b Fs(\))41 b(at)h(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b(t)p Fs(\))p Fo(;)g(t)p Fs(\).)74 b(Since)299 3866 y(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b(t)p Fs(\))p Fo(;)g(t)p Fs(\))28 b(is)d(the)i(p)s(oin)m (t)e(on)h(\010)1448 3880 y Fk(t)1477 3866 y Fs(\()p Fo(M)10 b Fs(\))27 b(nearest)g(to)g Fl(y)f Fs(it)g(follo)m(ws)f(that)i Fl(y)12 b Fm(\000)g Fs(\010\()p Fo(\025)p 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Fm(\000)20 b Fo(S)1938 2563 y Fk(v)1979 2549 y Fs(\))2014 2512 y Fe(\000)p Fn(1)2109 2549 y Fs(\()p Fo(\031)2196 2563 y Fk(F)2254 2549 y Fo(Y)40 b Fm(\000)20 b(h)p Fo(d)5 b(Y)g(;)15 b(v)s Fm(i)2705 2512 y Fn(#)2770 2549 y Fs(\))2805 2475 y Fi(\013)2868 2549 y Fm(\002)2511 2744 y Fo(p)p Fs(\()p Fo(F)33 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)3248 2758 y Fk(v)3288 2744 y Fs(\))p Fo(dv)s(;)-3442 b Fs(\(3.18\))0 2927 y(where)1313 3110 y Fo(\031)1365 3128 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))1543 3110 y Fs(:)26 b Fo(T)1647 3128 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))1800 3110 y Fl(R)1878 3072 y Fk(n)1951 3110 y Fm(!)25 b Fo(T)2120 3124 y Fk(x)2164 3110 y Fo(M)5 b(;)0 3293 y Fs(denotes)42 b(the)g(orthogonal)g(pro)5 b(jection)42 b(of)g Fo(T)1586 3311 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))1739 3293 y Fl(R)1817 3260 y Fk(n)1906 3293 y Fs(on)m(to)43 b Fo(T)2175 3307 y Fk(x)2219 3293 y Fo(M)10 b Fs(.)75 b(This)40 b(form)m(ula)h(ma)m(y)h(lo)s(ok)g(more)0 3406 y(complicated)30 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5267 y Fo(@)5 b(F)p 2647 5308 132 4 v 2647 5391 a(@)g(x)2752 5365 y Fk(l)2788 5173 y Fi(+)2862 5328 y Fo(:)-2887 b Fs(\(3.21\))p eop %%Page: 21 29 21 28 bop 3803 100 a Fs(21)299 490 y(De\014ne)30 b(the)h(n)m(um)m(b)s (ers)e([\()p Fo(I)f Fm(\000)20 b Fo(S)1379 504 y Fk(v)1419 490 y Fs(\))1454 457 y Fe(\000)p Fn(1)1549 490 y Fs(])1574 457 y Fk(m)1574 518 y(l)1671 490 y Fs(b)m(y)1373 733 y(\()p Fo(I)28 b Fm(\000)20 b Fo(S)1623 747 y Fk(v)1663 733 y Fs(\))1698 696 y Fe(\000)p Fn(1)1806 672 y Fo(@)5 b(F)p 1803 712 132 4 v 1803 796 a(@)g(x)1908 769 y Fk(l)1970 733 y Fs(=)24 b([\()p Fo(I)k Fm(\000)20 b Fo(S)2340 747 y Fk(v)2381 733 y Fs(\))2416 696 y Fe(\000)p Fn(1)2511 733 y Fs(])2536 696 y Fk(m)2536 756 y(l)2636 672 y Fo(@)5 b(F)p 2612 712 172 4 v 2612 796 a(@)g(x)2717 769 y Fk(m)2794 733 y Fo(;)299 955 y Fs(then)30 b(\(3)p Fo(:)p Fs(21\))j(giv)m(es)1153 1077 y Fi(\012)1195 1151 y Fm(h)p Fo(dX)r(;)15 b(v)s Fm(i)1476 1113 y Fn(#)1542 1151 y Fo(;)g Fs(\()p Fo(I)28 b Fm(\000)20 b Fo(S)1832 1165 y Fk(v)1872 1151 y Fs(\))1907 1113 y Fe(\000)p Fn(1)2002 1151 y Fm(h)p Fo(d)5 b(Y)g(;)15 b(v)s Fm(i)2269 1113 y Fn(#)2333 1077 y Fi(\013)999 1375 y Fs(=)83 b Fo(h)1205 1337 y Fk(ij)1265 1220 y Fi(*)1349 1313 y Fo(@)5 b(X)p 1349 1354 136 4 v 1350 1437 a(@)g(x)1455 1411 y Fk(i)1494 1375 y Fo(;)15 b(v)1581 1220 y Fi(+)1655 1375 y Fo(h)1707 1337 y Fk(k)r(l)1772 1220 y Fi(*)1866 1313 y Fo(@)5 b(Y)p 1855 1354 148 4 v 1855 1437 a(@)g(x)1960 1411 y Fk(k)2013 1375 y Fo(;)15 b(v)2100 1220 y Fi(+)q(*)2266 1313 y Fo(@)5 b(F)p 2257 1354 142 4 v 2257 1437 a(@)g(x)2362 1411 y Fk(j)2409 1375 y Fo(;)15 b Fs([\()p Fo(I)29 b Fm(\000)19 b Fo(S)2724 1389 y Fk(v)2765 1375 y Fs(\))2800 1337 y Fe(\000)p Fn(1)2895 1375 y Fs(])2920 1337 y Fk(m)2920 1398 y(l)3020 1313 y Fo(@)5 b(F)p 2997 1354 172 4 v 2997 1437 a(@)g(x)3102 1411 y Fk(m)3178 1220 y Fi(+)999 1681 y Fs(=)83 b Fo(h)1205 1643 y Fk(ij)1265 1525 y Fi(*)1349 1619 y Fo(@)5 b(X)p 1349 1660 136 4 v 1350 1743 a(@)g(x)1455 1717 y Fk(i)1494 1681 y Fo(;)15 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Fo(Y)46 b Fm(2)25 b Fo(T)2614 2090 y Fk(x)2658 2076 y Fo(M)41 b Fs(and)30 b(write)g(the)h(left)g(hand)f(side)299 2188 y(in)f(lo)s(cal)h(co)s (ordinates.)435 2463 y(Consider)g(a)i(smo)s(oth)f(v)-5 b(ariation)31 b(\010)c(:)g Fo(M)k Fm(\002)21 b Fs(\()p Fm(\000)p Fo(\017;)15 b Fs(+)p Fo(\017)p Fs(\))21 b Fm(\002)g Fs(\()p Fm(\000)p Fo(\017;)15 b Fs(+)p Fo(\017)p Fs(\))27 b Fm(!)g Fl(R)2942 2430 y Fk(n)3021 2463 y Fs(suc)m(h)k(that)h(\010\()p Fm(\001)p Fo(;)15 b Fs(0\))29 b(=)d Fo(F)13 b Fs(.)299 2576 y(Let)31 b(\010)528 2590 y Fk(st)615 2576 y Fs(=)25 b(\010\()p Fm(\001)p Fo(;)15 b(s;)g(t)p Fs(\))31 b(and)f(let)g Fo(\025)p Fs(\()p Fm(\001)p Fo(;)15 b(s;)g(t)p Fs(\))27 b(:)e(\012)g Fm(!)g Fo(M)41 b Fs(b)s(e)29 b(the)i(pro)5 b(jection)30 b(index)f(asso)s(ciated)i(with)e(\010)3747 2590 y Fk(st)3809 2576 y Fs(.)299 2757 y Fl(Theorem)34 b(3.4.2)46 b Fp(L)-5 b(et)40 b Fo(F)53 b Fp(b)-5 b(e)40 b(a)g(critic)-5 b(al)41 b(emb)-5 b(e)g(dding)41 b(of)f(the)g(distanc)-5 b(e)41 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b(\014elds)g(gener)-5 b(ate)g(d)45 b(by)e Fs(\010)p Fp(.)73 b(Mor)-5 b(e)g(over,)47 b(al)5 b(l)44 b(critic)-5 b(al)299 3539 y(emb)g(e)g(ddings)34 b(ar)-5 b(e)33 b(sadd)5 b(le)34 b(p)-5 b(oints)34 b(of)f Fo(E)1647 3554 y Fk(d)1688 3539 y Fp(.)435 3720 y Fs(Before)39 b(pro)m(ving)e(Theorem)g(3.4.2,)42 b(w)m(e)c(discuss)e(the)i(meaning)f(of)h(the)g(quan)m(tities)f Fo(P)3454 3734 y Fk(F)3550 3720 y Fs(and)g Fo(H)3810 3734 y Fk(F)3868 3720 y Fs(.)299 3832 y(W)-8 b(e)36 b(assume)e(that)i (for)e(almost)h(ev)m(ery)h Fo(x)c Fm(2)g Fo(M)45 b Fs(the)35 b(set)g Fo(supp)p Fs(\()p Fo(p)p Fs(\))23 b Fm(\\)g Fs(\012)2759 3851 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))2946 3832 y Fs(has)35 b(p)s(ositiv)m(e)f(measure)g(as)299 3945 y(a)f(subset)f(of)h(the)g (disk)e(\012)1181 3964 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))1334 3945 y Fs(.)48 b(Then,)32 b(the)h(function)f Fo(P)2248 3959 y Fk(F)2339 3945 y Fs(is)g(p)s(ositiv)m(e)g(almost)g(ev)m (erywhere)h(since)f(the)299 4058 y(determinan)m(t)e(det\()p Fo(I)e Fm(\000)20 b Fo(S)1188 4072 y Fk(v)1228 4058 y Fs(\))31 b(is)e(non)m(v)-5 b(anishing)28 b(in)h(\012.)41 b(Moreo)m(v)m(er,)32 b(in)m(tegration)f(of)f Fo(P)3217 4072 y Fk(F)3307 4058 y Fs(o)m(v)m(er)h Fo(M)41 b Fs(giv)m(es)610 4172 y Fi(Z)661 4378 y Fk(M)755 4296 y Fo(P)813 4310 y Fk(F)887 4296 y Fo(dv)s(ol)1052 4311 y Fk(h)1123 4296 y Fs(=)1219 4172 y Fi(Z)1270 4378 y Fk(M)1364 4172 y Fi(Z)1414 4378 y Fn(\012)1465 4392 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))1618 4296 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2478 4310 y Fk(v)2518 4296 y Fs(\))p Fo(dv)f(dv)s(ol)2828 4311 y Fk(h)2899 4296 y Fs(=)2995 4172 y Fi(Z)3046 4378 y Fn(\012)3116 4296 y Fo(p)p Fs(\()p Fl(x)p Fs(\))p Fo(d)p Fl(x)26 b Fs(=)f(1)p Fo(;)299 4560 y Fs(where)31 b(w)m(e)i(switc)m(hed)e(bac)m(k)h(from)g(normal)e(co)s (ordinates)i(to)h(Cartesian)e(co)s(ordinates.)45 b(Consequen)m(tly)-8 b(,)299 4673 y(the)30 b(function)g Fo(P)870 4687 y Fk(F)959 4673 y Fs(is)f(a)i(probabilit)m(y)d(densit)m(y)h(on)i Fo(M)10 b Fs(.)299 4853 y Fl(Prop)s(osition)36 b(3.4.3)46 b Fp(The)33 b(map)h Fo(H)1586 4867 y Fk(F)1677 4853 y Fp(satis\014es)f(the)g(pr)-5 b(op)g(erties:)378 5170 y(\(i\))46 b Fo(H)602 5184 y Fk(F)660 5170 y Fs(\()p Fo(X)r(;)15 b(Y)21 b Fs(\))26 b(=)f Fo(H)1119 5184 y Fk(F)1177 5170 y Fs(\()p Fo(Y)5 b(;)15 b(X)7 b Fs(\);)110 b Fo(X)r(;)15 b(Y)46 b Fm(2)25 b Fo(C)1936 5137 y Fe(1)2010 5170 y Fs(\()p Fo(M)5 b(;)15 b Ff(R)2257 5137 y Fk(n)2304 5170 y Fs(\))p Fp(.)351 5396 y(\(ii\))45 b Fo(H)602 5410 y Fk(F)660 5396 y Fs(\()p Fo(X)r(;)15 b(X)7 b Fs(\))28 b Fm(\025)d Fs(0)93 b Fp(and)h Fo(H)1504 5410 y Fk(F)1562 5396 y Fs(\()p Fo(X)r(;)15 b(X)7 b Fs(\))27 b(=)e(0)33 b Fp(if)f(and)i(only)g(if)e Fo(X)g Fm(2)25 b Fo(N)2849 5410 y Fk(F)2941 5396 y Fp(and)33 b Fm(r)3193 5363 y Fe(?)3252 5396 y Fo(X)g Fs(=)25 b(0)p Fp(.)p eop %%Page: 22 30 22 29 bop 0 100 a Fs(22)0 490 y Fp(Pr)-5 b(o)g(of)p Fs(.)69 b(Prop)s(ert)m(y)40 b(\(i\))f(follo)m(ws)g(from)g(the)g(fact)i(that)f (the)f(shap)s(e)g(op)s(erator)h Fo(S)2786 504 y Fk(v)2867 490 y Fs(:)g Fo(T)2985 504 y Fk(x)3029 490 y Fo(M)51 b Fm(!)40 b Fo(T)3352 504 y Fk(x)3396 490 y Fo(M)50 b Fs(is)0 603 y(selfadjoin)m(t.)39 b(T)-8 b(o)30 b(deriv)m(e)f(prop)s (ert)m(y)g(\(ii\),)g(observ)m(e)g(that)h(the)g(op)s(erator)f Fo(I)d Fm(\000)17 b Fo(S)2668 617 y Fk(v)2738 603 y Fs(is)29 b(p)s(ositiv)m(e)f(de\014nite)g(for)0 716 y(an)34 b(\012-regular)g(em)m (b)s(edding)e Fo(F)45 b Fm(2)32 b(E)1246 730 y Fn(\012)1335 716 y Fs(since)i(the)g(Jacobian)g(det)q(\()p Fo(I)c Fm(\000)22 b Fo(S)2487 730 y Fk(v)2528 716 y Fs(\))35 b(of)f(the)h(normal)e(co)s (ordinate)0 829 y(map)d(do)s(es)g(not)h(v)-5 b(anish)28 b(in)i(\012.)40 b(This)28 b(implies)g(that)j Fo(H)1876 843 y Fk(F)1934 829 y Fs(\()p Fo(X)r(;)15 b(X)7 b Fs(\))27 b Fm(\025)e Fs(0.)41 b(If)30 b Fo(H)2604 843 y Fk(F)2663 829 y Fs(\()p Fo(X)r(;)15 b(X)7 b Fs(\))27 b(=)e(0)30 b(then,)743 1025 y Fi(\012)786 1099 y Fo(\031)838 1113 y Fk(F)896 1099 y Fo(X)e Fm(\000)20 b(h)p Fo(dX)r(;)15 b(v)s Fm(i)1371 1061 y Fn(#)1436 1099 y Fo(;)g Fs(\()p Fo(I)28 b Fm(\000)20 b Fo(S)1726 1113 y Fk(v)1767 1099 y Fs(\))1802 1061 y Fe(\000)p Fn(1)1896 1099 y Fs(\()p Fo(\031)1983 1113 y Fk(F)2042 1099 y Fo(X)28 b Fm(\000)20 b(h)p Fo(dX)r(;)15 b(v)s Fm(i)2517 1061 y Fn(#)2582 1099 y Fs(\))2617 1025 y Fi(\013)2685 1099 y Fs(=)25 b(0)p Fo(;)0 1369 y Fs(for)37 b(all)f Fo(v)k Fm(2)c Fs(\012)526 1388 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))679 1369 y Fs(.)62 b(Since)36 b Fo(I)c Fm(\000)24 b Fo(S)1233 1383 y Fk(v)1311 1369 y Fs(is)36 b(p)s(ositiv)m(e)h(de\014nite,)h(w)m(e)f(ha)m(v)m(e)i Fo(\031)2519 1383 y Fk(F)2577 1369 y Fo(X)33 b Fm(\000)24 b(h)p Fo(dX)r(;)15 b(v)s Fm(i)3061 1336 y Fn(#)3163 1369 y Fs(=)36 b(0)i(for)f(all)0 1482 y Fo(v)32 b Fm(2)d Fo(N)239 1501 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))392 1482 y Fs(.)48 b(Set)33 b Fo(v)f Fs(=)d(0,)34 b(then)e Fo(\031)1165 1496 y Fk(F)1224 1482 y Fo(X)k Fs(=)29 b(0)k(whic)m(h)e(implies)f(that) k Fo(X)i Fm(2)29 b Fo(N)2560 1496 y Fk(F)2618 1482 y Fs(.)48 b(On)32 b(the)h(other)f(hand,)h(all)0 1595 y(\()p Fo(Y)5 b(;)15 b(v)s Fs(\))27 b Fm(2)d Fo(T)380 1609 y Fk(x)424 1595 y Fo(M)31 b Fm(\002)20 b Fs(\012)700 1613 y Fk(F)10 b Fn(\()p Fk(x)p Fn(\))883 1595 y Fs(is)29 b(suc)m(h)h(that)903 1879 y(0)c(=)1070 1806 y Fi(\012)1113 1879 y Fm(h)p Fo(dX)r(;)15 b(v)s Fm(i)1394 1842 y Fn(#)1459 1879 y Fo(;)g(Y)1572 1806 y Fi(\013)1640 1879 y Fs(=)25 b Fm(h)p Fo(dX)7 b Fs(\()p Fo(Y)22 b Fs(\))p Fo(;)15 b(v)s Fm(i)26 b Fs(=)f Fm(h)p Fo(v)s(;)15 b Fm(r)2487 1893 y Fk(Y)2549 1879 y Fo(X)7 b Fm(i)p Fo(;)0 2150 y Fs(whic)m(h)29 b(implies)f Fm(r)645 2117 y Fe(?)645 2177 y Fk(Y)705 2150 y Fo(X)33 b Fs(=)25 b(0)31 b(for)f(all)f Fo(Y)45 b Fm(2)25 b Fo(T)1487 2164 y Fk(x)1531 2150 y Fo(M)10 b Fs(.)41 b(Therefore,)30 b Fm(r)2208 2117 y Fe(?)2267 2150 y Fo(X)j Fs(=)25 b(0.)3280 2420 y Fl(Q.E.D.)136 2712 y Fs(W)-8 b(e)32 b(conclude)e(this)f(section)h(with)g(the)g(pro)s (of)g(of)g(Theorem)g(3.4.2.)43 b(By)31 b(\(3)p Fo(:)p Fs(16\),)391 2998 y Fo(@)444 2965 y Fn(2)484 2998 y Fo(E)551 3013 y Fk(d)592 2998 y Fs(\()p Fo(s;)15 b(t)p Fs(\))p 391 3039 387 4 v 494 3122 a Fo(@)5 b(s@)g(t)814 3060 y Fs(=)25 b Fm(\000)1012 2998 y Fo(@)p 991 3039 96 4 v 991 3122 a(@)5 b(s)1111 2936 y Fi(Z)1162 3142 y Fn(\012)1232 2904 y Fi(*)1305 3060 y Fl(y)21 b Fm(\000)f Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p Fo(;)2150 2998 y(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 2150 3039 682 4 v 2447 3122 a Fo(@)5 b(t)2841 2904 y Fi(+)2914 3060 y Fo(p)p Fs(\()p Fl(y)p Fs(\))p Fo(d)p Fl(y)q Fo(:)0 3401 y Fs(Let)626 3726 y Fo(g)s Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))26 b(=)f Fm(\000)1178 3664 y Fo(@)p 1157 3705 96 4 v 1157 3788 a(@)5 b(s)1262 3570 y Fi(*)1335 3726 y Fl(y)21 b Fm(\000)f Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p Fo(;)2180 3664 y(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 2180 3705 682 4 v 2477 3788 a Fo(@)5 b(t)2871 3570 y Fi(+)2944 3726 y Fo(:)0 4068 y Fs(Di\013eren)m(tiation)30 b(and)g(an)g(application)f(of)h(the)h(c)m(hain)f(rule)f(giv)m(e)400 4411 y Fo(g)s Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))84 b(=)966 4255 y Fi(*)1049 4349 y Fo(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 1049 4390 V 1323 4473 a Fo(@)5 b(x)1428 4447 y Fk(i)1750 4349 y Fo(@)g(\025)1856 4316 y Fk(i)1885 4349 y Fs(\()p Fl(y)p Fo(;)15 b(s;)g(t)p Fs(\))p 1750 4390 417 4 v 1910 4473 a Fo(@)5 b(s)2177 4411 y(;)2227 4349 y(@)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 2227 4390 682 4 v 2525 4473 a Fo(@)5 b(t)2918 4255 y Fi(+)966 4717 y Fs(+)1037 4561 y Fi(*)1120 4655 y Fo(@)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 1120 4696 V 1412 4779 a Fo(@)5 b(s)1811 4717 y(;)1861 4655 y(@)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 1861 4696 V 2159 4779 a Fo(@)5 b(t)2552 4561 y Fi(+)966 5023 y Fm(\000)1037 4867 y Fi(*)1110 5023 y Fl(y)20 b Fm(\000)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p Fo(;)1954 4961 y(@)2007 4928 y Fn(2)2047 4961 y Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)g(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 1954 5002 721 4 v 2205 5085 a Fo(@)5 b(x)2310 5059 y Fk(i)2338 5085 y Fo(@)g(t)2695 4961 y(@)g(\025)2801 4928 y Fk(i)2829 4961 y Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p 2695 5002 417 4 v 2855 5085 a Fo(@)5 b(s)3121 4867 y Fi(+)966 5328 y Fm(\000)1037 5173 y Fi(*)1110 5328 y Fl(y)20 b Fm(\000)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p Fo(;)1954 5267 y(@)2007 5234 y Fn(2)2047 5267 y Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)g(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 1954 5308 721 4 v 2223 5391 a Fo(@)5 b(s@)g(t)2685 5173 y Fi(+)2758 5328 y Fo(:)-2783 b Fs(\(3.23\))p eop %%Page: 23 31 23 30 bop 3803 100 a Fs(23)299 490 y(W)-8 b(e)31 b(compute)g Fo(g)s Fs(\()p Fl(y)q Fo(;)15 b Fs(0\).)42 b(Let)31 b Fo(Z)h Fs(=)25 b Fo(@)1554 457 y Fn(2)1594 490 y Fs(\010\()p Fm(\001)p Fo(;)15 b Fs(0\))p Fo(=@)5 b(s@)g(t)p Fs(.)42 b(Substituting)28 b(the)i(iden)m(tities)1421 664 y Fo(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 1421 705 541 4 v 1624 788 a Fo(@)5 b(x)1729 762 y Fk(i)2054 726 y Fs(=)2222 664 y Fo(@)g(F)p 2218 705 134 4 v 2218 788 a(@)g(x)2323 762 y Fk(i)2361 726 y Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))p Fo(;)-2421 b Fs(\(3.24\))1421 890 y Fo(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 1421 930 541 4 v 1643 1013 a Fo(@)5 b(s)2054 951 y Fs(=)83 b Fo(V)20 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))p Fo(;)-2341 b Fs(\(3.25\))1421 1115 y Fo(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 1421 1155 V 1648 1239 a Fo(@)5 b(t)2054 1176 y Fs(=)83 b Fo(W)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)i Fs(0\)\))p Fo(;)-2366 b Fs(\(3.26\))1046 1349 y Fo(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 1046 1389 V 1245 1473 a Fo(@)5 b(x)1350 1446 y Fk(j)1606 1349 y Fo(@)g(\025)1712 1316 y Fk(j)1749 1349 y Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p 1606 1389 355 4 v 1736 1473 a Fo(@)5 b(s)2054 1410 y Fs(=)2218 1349 y Fo(@)g(\025)p 2218 1389 107 4 v 2223 1473 a(@)g(s)2334 1410 y Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))p Fo(;)-2270 b Fs(\(3.27\))1007 1603 y Fo(@)1060 1570 y Fn(2)1099 1603 y Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 1007 1644 580 4 v 1182 1727 a Fo(@)5 b(x)1287 1701 y Fk(j)1324 1727 y Fo(@)g(t)1606 1603 y(@)g(\025)1712 1570 y Fk(j)1749 1603 y Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p 1606 1644 355 4 v 1736 1727 a Fo(@)5 b(s)2054 1665 y Fs(=)83 b Fo(dW)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)i Fs(0\)\))2687 1510 y Fi( )2771 1603 y Fo(@)5 b(\025)p 2771 1644 107 4 v 2776 1727 a(@)g(s)2887 1665 y Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))3098 1510 y Fi(!)3171 1665 y Fo(;)-2897 b Fs(\(3.28\))299 1930 y(in)m(to)30 b(\(3)p Fo(:)p Fs(23\))j(at)e Fo(s;)15 b(t)25 b Fs(=)g(0)31 b(giv)m(es)682 2196 y Fo(g)s Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))84 b(=)1177 2040 y Fi(*)1260 2134 y Fo(@)5 b(\025)p 1260 2175 V 1265 2258 a(@)g(s)1376 2196 y Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))21 b(+)f Fo(V)g Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))p Fo(;)g(W)e Fs(\()p Fo(\025)p Fs(\()p Fl(y)s Fo(;)i Fs(0\)\))2581 2040 y Fi(+)1177 2502 y Fm(\000)1248 2346 y Fi(*)1321 2502 y Fl(y)20 b Fm(\000)g Fo(F)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)i Fs(0\)\))p Fo(;)g(dW)e Fs(\()p Fo(\025)p Fs(\()p Fl(y)s Fo(;)i Fs(0\)\))2414 2346 y Fi( )2497 2440 y Fo(@)5 b(\025)p 2497 2481 V 2502 2564 a(@)g(s)2614 2502 y Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))2824 2346 y Fi(!)2917 2502 y Fs(+)20 b Fo(Z)7 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))3411 2346 y Fi(+)3486 2502 y Fo(:)-3212 b Fs(\(3.29\))299 2767 y(Since)29 b Fo(@)5 b(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(=@)5 b(s)32 b Fs(is)d(tangen)m(t)j(to)f Fo(M)10 b Fs(,)31 b(w)m(e)g(can)g(write)885 2877 y Fi(*)968 2971 y Fo(@)5 b(\025)p 968 3012 V 973 3095 a(@)g(s)1084 3033 y Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(;)g(W)e Fs(\()p Fo(\025)p Fs(\()p Fl(y)r Fo(;)i Fs(0\)\))1769 2877 y Fi(+)1869 3033 y Fs(=)1965 2877 y Fi(*)2048 2971 y Fo(@)5 b(\025)p 2048 3012 V 2053 3095 a(@)g(s)2164 3033 y Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(;)g(\031)2467 3051 y Fk(F)10 b Fn(\()p Fk(\025)p Fn(\()p Fh(y)r Fk(;)p Fn(0\)\))2776 3033 y Fo(W)j Fs(\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)i Fs(0\)\))3208 2877 y Fi(+)3283 3033 y Fo(:)-3009 b Fs(\(3.30\))299 3298 y(Substituting)28 b(\(3)p Fo(:)p Fs(30\))k(in)m(to)f(\(3)p Fo(:)p Fs(29\),)h(and)e(collecting)g(the)h(factors)g(of)g Fo(@)5 b(\025)p Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))p Fo(=@)5 b(s)32 b Fs(giv)m(e)485 3486 y Fo(g)s Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))84 b(=)f Fm(h)p Fo(V)20 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))p Fo(;)g(W)e Fs(\()p Fo(\025)p Fs(\()p Fl(y)s Fo(;)i Fs(0\)\))p Fm(i)980 3701 y Fs(+)1051 3546 y Fi(*)1124 3701 y Fo(\031)1176 3720 y Fk(F)10 b Fn(\()p Fk(\025)p Fn(\()p Fh(y)p Fk(;)p Fn(0\)\))1483 3701 y Fo(W)j Fs(\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)i Fs(0\)\))22 b Fm(\000)e(h)p Fo(dW)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)i Fs(0\)\))p Fo(;)g Fl(y)22 b Fm(\000)e Fo(F)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)i Fs(0\)\))p Fm(i)3191 3664 y Fn(#)3256 3701 y Fo(;)3307 3640 y(@)5 b(\025)p 3307 3680 V 3312 3764 a(@)g(s)3423 3701 y Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))3633 3546 y Fi(+)980 3916 y Fm(\000h)p Fl(y)20 b Fm(\000)g Fo(F)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)i Fs(0\)\))p Fo(;)g(Z)7 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)s Fo(;)15 b Fs(0\)\))p Fm(i)p Fo(:)-1863 b Fs(\(3.31\))299 4105 y(W)-8 b(e)31 b(need)g(a)f(form)m(ula)g(for)g Fo(@)5 b(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(=@)5 b(s)p Fs(.)42 b(Observ)m(e)30 b(that)846 4215 y Fi(*)919 4370 y Fl(y)21 b Fm(\000)f Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p Fo(;)1764 4309 y(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))p 1764 4349 682 4 v 2038 4432 a Fo(@)5 b(x)2143 4406 y Fk(i)2455 4215 y Fi(+)2553 4370 y Fs(=)25 b(0)p Fo(;)107 b(i)26 b Fs(=)f(1)p Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(m)p Fs(;)299 4644 y(iden)m(tically)30 b(in)h Fo(s)h Fs(and)g Fo(t)g Fs(since)g Fl(y)22 b Fm(\000)f Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b(s;)g(t)p Fs(\))p Fo(;)g(s;)g(t)p Fs(\))33 b(is)f(normal)f(to)i(\010)2804 4658 y Fk(st)2866 4644 y Fs(\()p Fo(M)10 b Fs(\))33 b(and)f Fo(@)5 b Fs(\010)3365 4658 y Fk(st)3427 4644 y Fo(=@)g(x)3577 4611 y Fk(i)3639 4644 y Fs(is)31 b(tan-)299 4757 y(gen)m(tial)f(to)h(\010)771 4771 y Fk(st)833 4757 y Fs(\()p Fo(M)10 b Fs(\).)42 b(Di\013eren)m (tiating)30 b(with)f(resp)s(ect)h(to)h Fo(s)f Fs(at)h Fo(s)25 b Fs(=)g(0)31 b(giv)m(es)785 5023 y Fm(\000)856 4867 y Fi(*)939 4961 y Fo(@)5 b Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 939 5002 541 4 v 1139 5085 a Fo(@)5 b(x)1244 5059 y Fk(j)1500 4961 y Fo(@)g(\025)1606 4928 y Fk(j)1642 4961 y Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p 1500 5002 355 4 v 1629 5085 a Fo(@)5 b(s)1884 5023 y Fs(+)1985 4961 y Fo(@)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 1985 5002 541 4 v 2207 5085 a Fo(@)5 b(s)2535 5023 y(;)2586 4961 y(@)g Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 2586 5002 V 2789 5085 a Fo(@)5 b(x)2894 5059 y Fk(i)3136 4867 y Fi(+)785 5328 y Fs(+)856 5173 y Fi(*)929 5328 y Fl(y)21 b Fm(\000)f Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p Fo(;)1633 5267 y(@)1686 5234 y Fn(2)1726 5267 y Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)g Fs(0\))p Fo(;)g Fs(0\))p 1633 5308 580 4 v 1785 5391 a Fo(@)5 b(x)1890 5365 y Fk(i)1919 5391 y Fo(@)g(x)2024 5365 y Fk(j)2233 5267 y Fo(@)g(\025)2339 5234 y Fk(j)2376 5267 y Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))p 2233 5308 355 4 v 2362 5391 a Fo(@)5 b(s)2617 5328 y Fs(+)2718 5267 y Fo(@)2771 5234 y Fn(2)2811 5267 y Fs(\010\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(;)g Fs(0\))p 2718 5308 580 4 v 2894 5391 a Fo(@)5 b(x)2999 5365 y Fk(i)3027 5391 y Fo(@)g(s)3308 5173 y Fi(+)3407 5328 y Fs(=)24 b(0)p Fo(:)p eop %%Page: 24 32 24 31 bop 0 100 a Fs(24)0 490 y(Using)30 b(the)g(iden)m(tities)f(\(3)p Fo(:)p Fs(24\))k(through)c(\(3)p Fo(:)p Fs(28\))k(giv)m(es)362 795 y Fm(\000)433 639 y Fi(*)516 733 y Fo(@)5 b(\025)p 516 774 107 4 v 521 857 a(@)g(s)632 795 y Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))21 b(+)f Fo(V)h Fs(\()p Fo(\025)p Fs(\()p Fl(y)p Fo(;)15 b Fs(0\)\))p Fo(;)1419 733 y(@)5 b(F)p 1414 774 134 4 v 1414 857 a(@)g(x)1519 831 y Fk(i)1558 795 y Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))1892 639 y Fi(+)362 1101 y Fs(+)433 945 y Fi(*)506 1101 y Fl(y)21 b Fm(\000)e Fo(F)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)i Fs(0\)\))p Fo(;)g Fm(r)1203 1104 y Fg(@)s(\025)p 1205 1116 73 3 v 1209 1157 a(@)s(s)1288 1131 y Fn(\()p Fh(y)p Fk(;)p Fn(0\))1459 1039 y Fo(@)5 b(F)p 1455 1080 134 4 v 1455 1163 a(@)g(x)1560 1137 y Fk(i)1598 1101 y Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))22 b(+)e Fo(dV)g Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))2499 945 y Fi( )2587 1039 y Fo(@)5 b(F)p 2583 1080 V 2583 1163 a(@)g(x)2688 1137 y Fk(i)2726 1101 y Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))3060 945 y Fi(!)q(+)3232 1101 y Fs(=)25 b(0)p Fo(:)-3398 b Fs(\(3.32\))0 1406 y(W)-8 b(e)32 b(use)e(\(2)p Fo(:)p Fs(2\))i(to)f(write)316 1555 y Fi(*)389 1711 y Fl(y)21 b Fm(\000)f Fo(F)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)i Fs(0\)\))p Fo(;)g Fm(r)1087 1714 y Fg(@)s(\025)p 1088 1726 73 3 v 1092 1767 a(@)s(s)1171 1741 y Fn(\()p Fh(y)p Fk(;)p Fn(0\))1342 1649 y Fo(@)5 b(F)p 1338 1690 134 4 v 1338 1773 a(@)g(x)1443 1747 y Fk(i)1481 1711 y Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))1815 1555 y Fi(+)1915 1711 y Fs(=)2011 1555 y Fi(*)2084 1711 y Fo(S)2140 1729 y Fh(y)p Fe(\000)p Fk(F)10 b Fn(\()p Fk(\025)p Fn(\()p Fh(y)p Fk(;)p Fn(0\)\))2555 1649 y Fo(@)5 b(\025)p 2555 1690 107 4 v 2560 1773 a(@)g(s)2672 1711 y Fs(\()p Fl(y)p Fo(;)15 b Fs(0\))p Fo(;)2938 1649 y(@)5 b(F)p 2934 1690 134 4 v 2934 1773 a(@)g(x)3039 1747 y Fk(i)3077 1711 y Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\)\))3411 1555 y Fi(+)3486 1711 y Fo(:)-3511 b Fs(\(3.33\))0 2024 y(Substituting)18 b(\(3)p Fo(:)p Fs(33\))23 b(in)m(to)e(\(3)p Fo(:)p Fs(32\),)k(and)20 b(collecting)h(the)g(factors)h(of)f Fo(@)5 b(\025)p Fs(\()p Fl(y)q Fo(;)15 b Fs(0\))p Fo(=@)5 b(s)22 b Fs(and)e Fo(@)5 b(F)13 b Fs(\()p Fo(\025)p Fs(\()p Fl(y)q Fo(;)i Fs(0\)\))p Fo(=@)5 b(x)3564 1991 y Fk(i)0 2137 y Fs(giv)m(e)337 2281 y Fi(*)410 2437 y Fs(\()p Fo(S)501 2455 y Fh(y)p Fe(\000)p Fk(F)10 b Fn(\()p Fk(\025)p Fn(\()p Fh(y)p Fk(;)p Fn(0\)\))926 2437 y Fm(\000)20 b Fo(I)7 b Fs(\))1109 2375 y Fo(@)e(\025)p 1109 2416 107 4 v 1114 2499 a(@)g(s)1226 2437 y 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b(H)3659 1509 y Fk(F)3747 1495 y Fs(and)299 1608 y Fo(\034)339 1622 y Fk(F)428 1608 y Fs(giv)m(e)935 1781 y Fo(@)988 1748 y Fn(2)1028 1781 y Fo(E)1095 1796 y Fk(d)p 935 1821 V 944 1905 a Fo(@)5 b(s@)g(t)1146 1683 y Fi(\014)1146 1738 y(\014)1146 1792 y(\014)1146 1847 y(\014)1146 1901 y(\014)1176 1960 y Fn(\()p Fk(s;t)p Fn(\)=0)1428 1842 y Fs(=)1524 1719 y Fi(Z)1575 1925 y Fk(M)1654 1842 y Fm(f)p Fo(P)1757 1856 y Fk(F)1816 1842 y Fm(h)p Fo(V)g(;)15 b(W)e Fm(i)21 b(\000)f Fo(H)2271 1856 y Fk(F)2329 1842 y Fs(\()p Fo(V)5 b(;)15 b(W)e Fs(\))21 b(+)f Fm(h)p Fo(\034)2783 1856 y Fk(F)2842 1842 y Fo(;)15 b(Z)7 b Fm(ig)p Fo(dv)s(ol)3196 1857 y Fk(h)3242 1842 y Fo(:)-2968 b Fs(\(3.35\))299 2103 y(If)30 b Fo(F)43 b Fs(is)30 b(a)g(critical)g(em)m(b)s(edding)e (of)j Fo(E)1592 2118 y Fk(d)1632 2103 y Fs(,)g(then)f Fo(\034)1935 2117 y Fk(F)2019 2103 y Fs(=)25 b(0)30 b(and)g(the)h(form) m(ula)e(\(3)p Fo(:)p Fs(22\))k(follo)m(ws.)435 2215 y(T)-8 b(o)37 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3252 y Fn(#)2343 3290 y Fm(j)2368 3252 y Fn(2)2408 3290 y Fo(p)p Fs(\()p Fo(F)34 b Fs(+)19 b Fo(v)s Fs(\))c(det)r(\()p Fo(I)27 b Fm(\000)20 b Fo(S)3145 3304 y Fk(v)3186 3290 y Fs(\))p Fo(dv)s(:)-3041 b Fs(\(3.36\))299 3520 y(No)m(w)34 b(w)m(e)h(giv)m(e)f(an)g(estimate)g (of)g(the)g(squared)f(norm)g(of)h(the)g(v)m(ector)i Fm(h)p Fo(dV)5 b(;)15 b(v)s Fm(i)2988 3487 y Fn(#)3083 3520 y Fm(2)31 b Fo(T)13 b(M)d Fs(.)51 b(Consider)32 b(an)299 3633 y(orthonormal)e(frame)g Fm(f)p Fo(e)1162 3647 y Fk(i)1191 3633 y Fm(g)h Fs(along)f Fo(F)13 b Fs(\()p Fo(M)d Fs(\).)42 b(W)-8 b(e)31 b(ha)m(v)m(e)1460 3865 y Fm(jh)p Fo(dV)5 b(;)15 b(v)s Fm(i)1747 3827 y Fn(#)1812 3865 y Fm(j)1837 3827 y Fn(2)1902 3865 y Fs(=)2032 3751 y Fk(m)1998 3779 y Fi(X)2006 3974 y Fk(i)p Fn(=1)2144 3791 y Fi(\012)2187 3865 y Fo(e)2229 3879 y Fk(i)2258 3865 y Fo(;)g Fm(h)p Fo(dV)5 b(;)15 b(v)s Fm(i)2560 3827 y Fn(#)2625 3791 y Fi(\013)2668 3814 y Fn(2)2707 3865 y Fo(:)299 4120 y Fs(Since)36 b(b)m(y)i(de\014nition)1088 4046 y 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Fs(Therefore,)31 b(c)m(ho)s(osing)g(a)g(v)-5 b(ariation)30 b(with)f(large)i Fm(jr)2096 5233 y Fe(?)2155 5266 y Fo(V)21 b Fm(j)31 b Fs(and)f(small)f Fm(j)p Fo(V)21 b Fm(j)31 b Fs(\(in)f(the)h Fo(L)3216 5233 y Fn(2)3286 5266 y Fs(sense\))g(in)e(\(3)p Fo(:)p Fs(22\))299 5379 y(leads)h(to)h(a)g(negativ)m(e)g(second)g(deriv)-5 b(ativ)m(e.)p eop %%Page: 26 34 26 33 bop 1583 805 a Fq(Chapter)34 b(4)192 1005 y Fj(EXTREMAL)i(PR)m (OPER)-9 b(TIES)35 b(OF)j(THE)e(ENER)m(GY)g(FUNCTIONAL)136 1284 y Fs(In)j(this)g(c)m(hapter)h(w)m(e)g(la)m(y)g(the)g(ground)f(for) g(the)h(study)f(of)g(the)h(general)g(\014tting)f(problem.)67 b(Our)0 1396 y(main)25 b(concerns)h(are)h(the)f(v)-5 b(ariational)25 b(prop)s(erties)f(of)i Fo(E)32 b Fs(on)26 b(the)g(class)g(of)g(\012-regular)f(em)m(b)s(eddings.)38 b(W)-8 b(e)0 1509 y(c)m(haracterize)31 b(the)f(critical)e(p)s(oin)m(ts) g(of)i Fo(E)k Fs(in)28 b(terms)i(of)f(harmonic)f(maps)h(and)g(the)h (extrinsic)d(geometry)0 1622 y(of)41 b(the)g(em)m(b)s(edding)e Fo(F)56 b Fs(:)43 b Fo(M)52 b Fm(!)43 b Fl(R)1285 1589 y Fk(n)1332 1622 y Fs(.)72 b(W)-8 b(e)42 b(also)f(in)m(v)m(estigate)h (conditions)d(under)g(whic)m(h)h(a)h(critical)0 1735 y(em)m(b)s(edding)29 b(yields)f(a)j(lo)s(cal)f(minim)m(um.)0 1982 y Fc(4.1)105 b(Regularit)m(y)36 b(prop)s(erties)f(of)g Fo(E)0 2185 y Fs(In)d(this)g(section)h(w)m(e)h(con)m(tin)m(ue)f(with)f (the)h(analysis)e(of)j(the)f(regularit)m(y)f(of)h Fo(E)38 b Fs(started)c(in)e(Section)g(3.2.)0 2298 y(W)-8 b(e)41 b(will)d(sho)m(w)i(that)h(the)f(harmonic)g(functional)e Fo(E)1869 2313 y Fk(h)1954 2298 y Fs(is)i(a)g Fo(C)2213 2265 y Fe(1)2328 2298 y Fs(map)f(de\014ned)g(on)i(an)f(appropriate)0 2411 y(Sob)s(olev)29 b(space.)136 2524 y(F)-8 b(or)31 b Fo(k)e(>)c(m)p Fs(,)30 b(consider)g(the)g(map)g Fo(E)h Fs(:)25 b Fm(E)1523 2486 y Fk(k)r Fn(+2)1515 2553 y(\012)1676 2524 y Fm(\002)20 b Fo(C)1839 2491 y Fk(k)r Fn(+3)1972 2524 y Fs(\(\012\))g Fm(\002)g Fl(R)26 b Fm(!)f Fl(R)31 b Fs(giv)m(en)f(b)m(y)1156 2711 y Fo(E)5 b Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))27 b(=)e Fo(E)1725 2726 y Fk(d)1766 2711 y Fs(\()p Fo(F)s(;)15 b(p)p Fs(\))21 b(+)f Fo(k)s(E)2212 2726 y Fk(h)2257 2711 y Fs(\()p Fo(F)13 b Fs(\))p Fo(;)-2423 b Fs(\(4.1\))0 2898 y(where)38 b Fo(E)338 2913 y Fk(d)417 2898 y Fs(is)f(de\014ned)g(in)g(\(3)p Fo(:)p Fs(12\))k(and)c Fo(E)1473 2913 y Fk(h)1557 2898 y Fs(is)g(the)i(harmonic)e(functional)f (\(1)p Fo(:)p Fs(2\).)67 b(Our)37 b(goal)i(in)e(this)0 3010 y(section)31 b(is)e(to)i(pro)m(v)m(e)g(the)g(follo)m(wing)e (theorem.)0 3184 y Fl(Theorem)34 b(4.1.1)46 b Fp(L)-5 b(et)33 b Fo(k)c(>)c(m)p Fp(.)41 b(Then)33 b Fo(E)38 b Fp(is)33 b(a)g Fo(C)1773 3151 y Fn(2)1811 3184 y Fp(-map.)0 3357 y(Pr)-5 b(o)g(of)p Fs(.)43 b(W)-8 b(e)32 b(sho)m(w)f(that)g Fo(E)925 3372 y Fk(h)996 3357 y Fs(:)26 b(\000)1104 3324 y Fk(k)r Fn(+2)1236 3357 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1484 3324 y Fk(n)1531 3357 y Fs(\))26 b Fm(!)f Fl(R)32 b Fs(is)d(a)i(smo)s (oth)g(map.)41 b(Then,)30 b(from)h(Theorem)f(3.2.1)0 3470 y(it)g(will)e(follo)m(w)h(that)i Fo(E)36 b Fs(is)29 b Fo(C)988 3437 y Fn(2)1027 3470 y Fs(.)136 3582 y(Consider)39 b(a)h(lo)s(cal)g(orthonormal)f(frame)i Fm(f)p Fo(e)1719 3549 y Fk(M)1719 3608 y(i)1798 3582 y Fm(g)i(\032)e Fo(T)13 b(M)50 b Fs(with)39 b(resp)s(ect)h(to)h(the)f(metric)g Fo(g)s Fs(.)71 b(Let)0 3695 y Fo(F)s(;)15 b(V)46 b Fm(2)25 b Fs(\000)343 3662 y Fk(k)r Fn(+2)475 3695 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)723 3662 y Fk(n)770 3695 y Fs(\).)41 b(W)-8 b(e)31 b(ha)m(v)m(e)511 3800 y Fi(\014)511 3855 y(\014)511 3909 y(\014)511 3964 y(\014)511 4018 y(\014)551 3898 y Fo(E)618 3913 y Fk(h)663 3898 y Fs(\()p Fo(F)j Fs(+)20 b Fo(tV)g Fs(\))g Fm(\000)g Fo(E)1200 3913 y Fk(h)1245 3898 y Fs(\()p Fo(F)13 b Fs(\))p 551 3938 836 4 v 953 4022 a Fo(t)1417 3959 y Fm(\000)1508 3836 y Fi(Z)1558 4042 y Fk(M)1653 3959 y Fo(\016)1696 3922 y Fk(ij)1757 3959 y Fm(h)p Fo(dF)g Fs(\()p Fo(e)1987 3922 y Fk(M)1987 3982 y(i)2067 3959 y Fs(\))p Fo(;)i(dV)22 b Fs(\()p Fo(e)2341 3922 y Fk(M)2341 3982 y(j)2420 3959 y Fs(\))p Fm(i)p Fo(dv)s(ol)2655 3973 y Fk(g)2697 3800 y Fi(\014)2697 3855 y(\014)2697 3909 y(\014)2697 3964 y(\014)2697 4018 y(\014)357 4265 y Fs(=)511 4106 y Fi(\014)511 4161 y(\014)511 4215 y(\014)511 4270 y(\014)511 4324 y(\014)561 4156 y Fn(1)p 561 4171 36 4 v 561 4223 a(2)622 4118 y Fi(R)664 4223 y Fk(M)759 4191 y Fo(\016)802 4158 y Fk(ij)863 4191 y Fm(fh)p Fo(d)p Fs(\()p Fo(F)35 b Fs(+)20 b Fo(tV)g Fs(\)\()p Fo(e)1427 4158 y Fk(M)1427 4217 y(i)1507 4191 y Fs(\))p Fo(;)15 b(d)p Fs(\()p Fo(F)35 b Fs(+)19 b Fo(tV)h Fs(\)\()p Fo(e)2065 4158 y Fk(M)2065 4217 y(j)2146 4191 y Fs(\))p Fm(i)g(\000)g(h)p Fo(dF)13 b Fs(\()p Fo(e)2557 4158 y Fk(M)2557 4217 y(i)2638 4191 y Fs(\))p Fo(;)i(dF)e Fs(\()p Fo(e)2908 4158 y Fk(M)2908 4217 y(j)2989 4191 y Fs(\))p Fm(ig)p Fo(dv)s(ol)3269 4205 y Fk(g)p 551 4244 2760 4 v 1915 4328 a Fo(t)602 4571 y Fm(\000)688 4447 y Fi(Z)738 4654 y Fk(M)832 4571 y Fo(\016)875 4534 y Fk(ij)937 4571 y Fm(h)p Fo(dF)g Fs(\()p Fo(e)1167 4534 y Fk(M)1167 4594 y(i)1247 4571 y Fs(\))p Fo(;)i(dV)21 b Fs(\()p Fo(e)1520 4534 y Fk(M)1520 4594 y(j)1600 4571 y Fs(\))p Fm(i)p Fo(dv)s(ol)1835 4585 y Fk(g)1877 4412 y Fi(\014)1877 4467 y(\014)1877 4521 y(\014)1877 4576 y(\014)1877 4630 y(\014)357 4842 y Fs(=)527 4780 y Fo(t)p 521 4821 46 4 v 521 4904 a Fs(2)591 4718 y Fi(Z)642 4924 y Fk(M)736 4842 y Fo(\016)779 4804 y Fk(ij)840 4842 y Fm(h)p Fo(dV)g Fs(\()p Fo(e)1073 4804 y Fk(M)1073 4864 y(i)1153 4842 y Fs(\))p Fo(;)15 b(dV)21 b Fs(\()p Fo(e)1426 4804 y Fk(M)1426 4864 y(j)1506 4842 y Fs(\))p Fm(i)p Fo(dv)s(ol)1741 4856 y Fk(g)1783 4842 y Fo(:)0 5070 y Fs(Th)m(us,)523 5328 y(lim)520 5386 y Fk(t)p Fe(!)p Fn(0)666 5169 y Fi(\014)666 5224 y(\014)666 5278 y(\014)666 5333 y(\014)666 5388 y(\014)707 5267 y Fo(E)774 5282 y Fk(h)819 5267 y Fs(\()p Fo(F)33 b Fs(+)20 b Fo(tV)g Fs(\))h Fm(\000)f Fo(E)1356 5282 y Fk(h)1401 5267 y Fs(\()p Fo(F)13 b Fs(\))p 707 5308 836 4 v 1108 5391 a Fo(t)1573 5328 y Fm(\000)1664 5205 y Fi(Z)1714 5411 y Fk(M)1808 5328 y Fo(\016)1851 5291 y Fk(ij)1913 5328 y Fm(h)p Fo(dF)g Fs(\()p Fo(e)2143 5291 y Fk(M)2143 5351 y(i)2223 5328 y Fs(\))p Fo(;)i(dV)21 b Fs(\()p Fo(e)2496 5291 y Fk(M)2496 5351 y(j)2576 5328 y Fs(\))p Fm(i)p Fo(dv)s(ol)2811 5342 y Fk(g)2853 5169 y Fi(\014)2853 5224 y(\014)2853 5278 y(\014)2853 5333 y(\014)2853 5388 y(\014)2908 5328 y Fs(=)k(0)p Fo(:)p eop %%Page: 27 35 27 34 bop 3803 100 a Fs(27)299 490 y(Then,)30 b(the)g(G^)-45 b(ateaux)32 b(deriv)-5 b(ativ)m(e)30 b Fo(\016)s(E)1612 505 y Fk(h)1658 490 y Fs(\()p Fo(F)13 b Fs(\))26 b Fm(2)e Fs(\000)1967 457 y Fk(k)r Fn(+2)2100 490 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2347 457 y Fk(n)2394 490 y Fs(\))2429 457 y Fe(\003)2499 490 y Fs(of)31 b Fo(E)2670 505 y Fk(h)2745 490 y Fs(at)g Fo(F)44 b Fs(is)29 b(giv)m(en)h(b)m(y)1210 747 y Fo(\016)s(E)1320 762 y Fk(h)1366 747 y Fs(\()p Fo(F)13 b Fs(\))21 b Fm(\001)g Fo(V)45 b Fs(=)1768 624 y Fi(Z)1818 830 y Fk(M)1913 747 y Fo(\016)1956 710 y Fk(ij)2017 747 y Fm(h)p Fo(dF)13 b Fs(\()p Fo(e)2247 710 y Fk(M)2247 770 y(i)2327 747 y Fs(\))p Fo(;)i(dV)22 b Fs(\()p Fo(e)2601 710 y Fk(M)2601 770 y(j)2680 747 y Fs(\))p Fm(i)p Fo(dv)s(ol)2915 761 y Fk(g)2957 747 y Fo(:)-2683 b Fs(\(4.2\))299 1020 y(W)-8 b(e)31 b(no)m(w)g(pro)m(v)m (e)g(that)g(this)e(is)h(con)m(tin)m(uous)g(as)g(a)h(map)f Fo(\016)s(E)2310 1035 y Fk(h)2381 1020 y Fs(:)c(\000)2489 987 y Fk(k)r Fn(+2)2621 1020 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2868 987 y Fk(n)2915 1020 y Fs(\))26 b Fm(!)f Fs(\000)3149 987 y Fk(k)r Fn(+2)3282 1020 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3529 987 y Fk(n)3576 1020 y Fs(\))3611 987 y Fe(\003)3651 1020 y Fs(.)40 b(This)299 1133 y(will)28 b(imply)g(that)j Fo(E)990 1148 y Fk(h)1065 1133 y Fs(is)e(a)i Fo(C)1304 1100 y Fn(1)1343 1133 y Fs(-map)f(\([12)r(])g(p.37\).)42 b(F)-8 b(or)31 b Fo(F)2271 1147 y Fn(1)2311 1133 y Fo(;)15 b(F)2409 1147 y Fn(2)2475 1133 y Fm(2)24 b Fs(\000)2617 1100 y Fk(k)r Fn(+2)2750 1133 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2997 1100 y Fk(n)3044 1133 y Fs(\),)31 b(w)m(e)g(ha)m(v)m(e)905 1349 y Fm(j)p Fo(\016)s(E)1040 1364 y Fk(h)1086 1349 y Fs(\()p Fo(F)1179 1363 y Fn(1)1220 1349 y Fs(\))20 b Fm(\001)h Fo(V)40 b Fm(\000)20 b Fo(\016)s(E)1615 1364 y Fk(h)1661 1349 y Fs(\()p Fo(F)1754 1363 y Fn(2)1794 1349 y Fs(\))h Fm(\001)f Fo(V)g Fm(j)752 1564 y Fs(=)905 1405 y Fi(\014)905 1459 y(\014)905 1514 y(\014)905 1568 y(\014)905 1623 y(\014)951 1440 y(Z)1001 1646 y Fk(M)1095 1564 y Fo(\016)1138 1526 y Fk(ij)1200 1564 y Fm(h)p Fo(dF)1340 1578 y Fn(1)1380 1564 y Fs(\()p Fo(e)1457 1526 y Fk(M)1457 1586 y(i)1537 1564 y Fs(\))p Fo(;)15 b(dV)21 b Fs(\()p Fo(e)1810 1526 y Fk(M)1810 1586 y(j)1890 1564 y Fs(\))p Fm(i)p Fo(dv)s(ol)2125 1578 y Fk(g)2187 1564 y Fm(\000)2278 1440 y Fi(Z)2328 1646 y Fk(M)2422 1564 y Fo(\016)2465 1526 y Fk(ij)2527 1564 y Fm(h)p Fo(dF)2667 1578 y Fn(2)2707 1564 y Fs(\()p Fo(e)2784 1526 y Fk(M)2784 1586 y(i)2864 1564 y Fs(\))p Fo(;)15 b(dV)21 b Fs(\()p Fo(e)3137 1526 y Fk(M)3137 1586 y(j)3217 1564 y Fs(\))p Fm(i)p Fo(dv)s(ol)3452 1578 y Fk(g)3494 1405 y Fi(\014)3494 1459 y(\014)3494 1514 y(\014)3494 1568 y(\014)3494 1623 y(\014)752 1779 y Fm(\024)82 b(jj)p Fo(F)1013 1793 y Fn(1)1074 1779 y Fm(\000)20 b Fo(F)1223 1793 y Fn(2)1263 1779 y Fm(jj)1313 1793 y Fn(1)1353 1779 y Fm(jj)p Fo(V)g Fm(jj)1526 1793 y Fn(1)1566 1779 y Fo(:)299 1995 y Fs(Th)m(us)27 b Fm(jj)p Fo(\016)s(E)687 2010 y Fk(h)733 1995 y Fs(\()p Fo(F)826 2009 y Fn(1)867 1995 y Fs(\))16 b Fm(\000)g Fo(\016)s(E)1115 2010 y Fk(h)1160 1995 y Fs(\()p Fo(F)1253 2009 y Fn(2)1294 1995 y Fs(\))p Fm(jj)1379 2017 y Fn(\000)1423 1998 y Fg(k)q Fb(+2)1539 2017 y Fn(\()p Fk(M)s(;)p Fh(R)1717 1998 y Fg(n)1760 2017 y Fn(\))1787 1998 y Fa(\003)1853 1995 y Fm(\024)25 b(jj)p Fo(F)2057 2009 y Fn(1)2113 1995 y Fm(\000)16 b Fo(F)2258 2009 y Fn(2)2298 1995 y Fm(jj)2348 2010 y Fk(k)r Fn(+2)2481 1995 y Fs(;)29 b(i.e.,)g Fo(E)2772 2010 y Fk(h)2845 1995 y Fs(is)f(a)g Fo(C)3080 1962 y Fn(1)3119 1995 y Fs(-map.)40 b(By)29 b(\(4)p Fo(:)p Fs(2\),)h(the)299 2108 y(map)g Fo(\016)s(E)611 2123 y Fk(h)687 2108 y Fs(is)f(linear,)g (therefore)i(smo)s(oth.)41 b(This)28 b(pro)m(v)m(es)j(that)g Fo(E)2547 2123 y Fk(h)2623 2108 y Fs(is)e(smo)s(oth.)299 2379 y Fc(4.2)105 b(First)35 b(v)-6 b(ariation)35 b(of)g(E)299 2590 y Fs(In)26 b(this)f(section)i(w)m(e)g(c)m(haracterize)h(the)f (critical)e(em)m(b)s(eddings)g(of)i(the)f(functional)f Fo(E)32 b Fs(b)m(y)27 b(computing)e(its)299 2703 y(\014rst)30 b(v)-5 b(ariational)29 b(deriv)-5 b(ativ)m(e)30 b(at)h(an)f (\012-regular)g(em)m(b)s(edding)e Fo(F)13 b Fs(.)435 2819 y(Let)37 b(\001)680 2833 y Fk(g)757 2819 y Fs(denote)g(the)g (Laplace-Beltrami)g(op)s(erator)g(relativ)m(e)g(to)g(the)g(metric)g Fo(g)j Fs(in)c Fo(M)10 b Fs(.)59 b(In)36 b(lo)s(cal)299 2932 y(co)s(ordinates,)1487 3220 y(\001)1563 3234 y Fk(g)1603 3220 y Fo(f)f Fs(=)25 b Fm(\000)1898 3159 y Fs(1)p 1860 3199 123 4 v 1860 3226 a Fm(p)p 1935 3226 47 4 v 1935 3283 a Fo(g)2042 3159 y(@)p 2002 3199 134 4 v 2002 3283 a(@)5 b(x)2107 3256 y Fk(i)2145 3065 y Fi( )2217 3159 y Fm(p)p 2293 3159 47 4 v 61 x Fo(g)s(g)2385 3183 y Fk(ij)2474 3159 y Fo(@)g(f)p 2456 3199 142 4 v 2456 3283 a(@)g(x)2561 3256 y Fk(j)2608 3065 y Fi(!)2680 3220 y Fo(:)-2406 b Fs(\(4.3\))299 3523 y(Consider)28 b(a)i(lo)s(cal)f(orthonormal)g(frame) h Fm(f)p Fo(e)1829 3490 y Fk(M)1829 3549 y(i)1909 3523 y Fm(g)25 b(\032)g Fo(T)13 b(M)40 b Fs(with)28 b(resp)s(ect)i(to)h(the) f(metric)f Fo(g)k Fs(and)d(let)f Fo(h)h Fs(b)s(e)299 3636 y(the)g(Euclidean)f(metric)h(induced)e(on)j Fo(M)10 b Fs(.)299 3843 y Fl(Theorem)34 b(4.2.1)46 b Fp(L)-5 b(et)37 b Fo(F)49 b Fp(b)-5 b(e)37 b(an)g Fs(\012)p Fp(-r)-5 b(e)g(gular)37 b(emb)-5 b(e)g(dding)38 b(and)f(let)g Fs(\010)f Fp(b)-5 b(e)36 b(a)h(smo)-5 b(oth)39 b(variation)f(of)f Fo(F)13 b Fp(.)299 3956 y(Then,)990 4183 y Fo(dE)5 b Fs([\010\()p Fm(\001)p Fo(;)15 b(t)p Fs(\)])p 990 4223 406 4 v 1152 4306 a Fo(dt)1405 4085 y Fi(\014)1405 4139 y(\014)1405 4194 y(\014)1405 4249 y(\014)1405 4303 y(\014)1435 4362 y Fk(t)p Fn(=0)1580 4244 y Fs(=)1676 4120 y Fi(Z)1727 4326 y Fk(M)1806 4244 y Fm(h)p Fo(\034)1881 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10371 9928 l 10350 9938 l 10327 9947 l 10304 9957 l 10280 9966 l 10257 9975 l 10232 9984 l 10207 9993 l 10182 10002 l 10156 10010 l 10129 10018 l 10103 10026 l 10076 10034 l 10048 10041 l 10021 10048 l 9994 10055 l 9966 10061 l 9940 10067 l 9913 10072 l 9887 10077 l 9861 10082 l 9836 10086 l 9811 10090 l 9786 10094 l 9762 10097 l 9738 10100 l 9713 10103 l 9688 10105 l 9663 10108 l 9637 10110 l 9611 10112 l 9584 10113 l 9557 10115 l 9529 10116 l 9500 10117 l 9471 10118 l 9442 10118 l 9412 10118 l 9383 10118 l 9354 10118 l 9325 10117 l 9296 10116 l 9268 10115 l 9241 10113 l 9214 10112 l 9188 10110 l 9162 10108 l 9137 10105 l 9112 10103 l 9088 10100 l 9063 10097 l 9039 10094 l 9014 10090 l 8989 10086 l 8964 10082 l 8938 10077 l 8912 10071 l 8885 10065 l 8858 10059 l 8830 10052 l 8803 10044 l 8775 10036 l 8747 10027 l 8720 10018 l 8692 10008 l 8665 9998 l 8638 9987 l 8612 9976 l 8586 9965 l 8561 9953 l 8536 9940 l 8511 9927 l 8487 9914 l 8463 9900 l 8442 9887 l 8420 9874 l 8399 9861 l 8377 9846 l 8355 9831 l 8333 9815 l 8310 9798 l 8287 9781 l 8263 9763 l 8239 9744 l 8215 9725 l 8191 9704 l 8167 9684 l 8143 9663 l 8119 9642 l 8095 9620 l 8072 9598 l 8049 9576 l 8026 9554 l 8004 9532 l 7983 9511 l 7962 9489 l 7942 9468 l 7922 9446 l 7903 9425 l 7885 9404 l 7867 9383 l 7850 9363 l 7832 9340 l 7814 9317 l 7796 9294 l 7778 9271 l 7760 9247 l 7742 9222 l 7725 9197 l 7707 9172 l 7689 9147 l 7672 9121 l 7654 9095 l 7637 9069 l 7619 9043 l 7602 9017 l 7586 8991 l 7569 8966 l 7553 8942 l 7538 8918 l 7522 8894 l 7508 8871 l 7493 8850 l 7479 8828 l 7465 8808 l 7452 8788 l 7438 8769 l 7425 8750 l 7409 8728 l 7394 8707 l 7378 8686 l 7361 8665 l 7345 8644 l 7328 8624 l 7310 8603 l 7293 8583 l 7275 8563 l 7257 8544 l 7239 8525 l 7221 8506 l 7204 8488 l 7186 8471 l 7169 8455 l 7153 8439 l 7137 8425 l 7121 8411 l 7106 8398 l 7091 8385 l 7077 8374 l 7063 8363 l 7045 8349 l 7028 8337 l 7011 8324 l 6993 8312 l 6975 8300 l 6956 8289 l 6936 8277 l 6917 8266 l 6897 8255 l 6876 8245 l 6856 8235 l 6835 8225 l 6815 8216 l 6794 8207 l 6774 8199 l 6754 8190 l 6733 8183 l 6713 8175 l 6695 8169 l 6677 8162 l 6658 8156 l 6639 8150 l 6619 8143 l 6598 8136 l 6576 8130 l 6554 8123 l 6531 8117 l 6508 8110 l 6484 8103 l 6460 8097 l 6436 8091 l 6413 8085 l 6389 8080 l 6366 8074 l 6343 8070 l 6321 8065 l 6300 8061 l 6279 8057 l 6258 8053 l 6238 8050 l 6217 8047 l 6197 8044 l 6176 8042 l 6155 8039 l 6134 8037 l 6112 8035 l 6090 8033 l 6068 8031 l 6045 8030 l 6022 8028 l 5998 8027 l 5975 8025 l 5951 8024 l 5928 8023 l 5905 8022 l 5882 8020 l 5860 8019 l 5837 8018 l 5815 8017 l 5793 8015 l 5772 8014 l 5750 8013 l 5728 8011 l 5705 8009 l 5682 8007 l 5659 8005 l 5634 8003 l 5609 8001 l 5583 7998 l 5556 7996 l 5529 7993 l 5501 7990 l 5473 7987 l 5445 7985 l 5417 7982 l 5390 7979 l 5363 7977 l 5336 7974 l 5310 7972 l 5284 7970 l 5259 7968 l 5235 7966 l 5211 7964 l 5188 7963 l 5164 7961 l 5141 7960 l 5117 7958 l 5093 7957 l 5069 7956 l 5044 7955 l 5019 7954 l 4993 7954 l 4967 7953 l 4942 7952 l 4916 7952 l 4890 7951 l 4865 7951 l 4840 7951 l 4816 7950 l 4792 7950 l 4769 7950 l 4747 7950 l 4725 7950 l 4704 7950 l 4683 7950 l 4663 7950 l 4642 7950 l 4622 7950 l 4601 7950 l 4580 7950 l 4559 7950 l 4537 7949 l 4515 7949 l 4492 7949 l 4469 7948 l 4445 7947 l 4422 7947 l 4398 7946 l 4374 7944 l 4350 7943 l 4326 7941 l 4302 7940 l 4278 7938 l 4255 7936 l 4232 7933 l 4209 7931 l 4186 7928 l 4163 7925 l 4141 7922 l 4119 7919 l 4096 7915 l 4072 7911 l 4048 7907 l 4023 7903 l 3997 7898 l 3971 7894 l 3944 7888 l 3917 7883 l 3889 7877 l 3861 7872 l 3833 7866 l 3805 7860 l 3777 7854 l 3749 7847 l 3722 7841 l 3696 7835 l 3670 7829 l 3645 7823 l 3620 7817 l 3597 7812 l 3573 7806 l 3550 7800 l 3525 7794 l 3500 7787 l 3475 7781 l 3450 7774 l 3425 7768 l 3399 7761 l 3373 7754 l 3347 7747 l 3320 7740 l 3294 7732 l 3268 7725 l 3242 7718 l 3216 7710 l 3191 7703 l 3166 7696 l 3142 7689 l 3119 7682 l 3097 7676 l 3075 7669 l 3054 7663 l 3033 7656 l 3013 7650 l 2992 7644 l 2972 7637 l 2951 7631 l 2930 7624 l 2909 7618 l 2887 7611 l 2865 7604 l 2842 7596 l 2819 7589 l 2795 7581 l 2772 7573 l 2748 7565 l 2724 7558 l 2700 7550 l 2676 7542 l 2652 7534 l 2628 7526 l 2605 7518 l 2582 7511 l 2559 7503 l 2536 7495 l 2513 7488 l 2492 7481 l 2472 7474 l 2451 7467 l 2430 7460 l 2407 7452 l 2385 7444 l 2361 7436 l 2337 7427 l 2312 7418 l 2287 7408 l 2261 7398 l 2235 7388 l 2209 7377 l 2183 7366 l 2156 7355 l 2130 7343 l 2104 7331 l 2078 7319 l 2053 7306 l 2028 7294 l 2003 7281 l 1979 7268 l 1956 7254 l 1933 7241 l 1910 7227 l 1888 7213 l 1867 7199 l 1846 7185 l 1825 7170 l 1804 7154 l 1782 7138 l 1761 7121 l 1739 7104 l 1717 7086 l 1694 7067 l 1672 7047 l 1649 7027 l 1627 7006 l 1604 6985 l 1582 6963 l 1560 6941 l 1538 6918 l 1516 6896 l 1495 6873 l 1474 6850 l 1454 6828 l 1435 6805 l 1416 6783 l 1397 6760 l 1379 6738 l 1362 6716 l 1345 6694 l 1329 6672 l 1313 6650 l 1297 6628 l 1281 6605 l 1265 6582 l 1250 6559 l 1234 6535 l 1219 6510 l 1203 6485 l 1188 6459 l 1172 6432 l 1157 6405 l 1141 6377 l 1126 6348 l 1111 6319 l 1096 6290 l 1082 6260 l 1067 6230 l 1053 6200 l 1039 6170 l 1026 6140 l 1013 6110 l 1001 6080 l 989 6050 l 977 6021 l 966 5992 l 956 5963 l 945 5933 l 935 5904 l 925 5875 l 916 5849 l 908 5823 l 900 5796 l 891 5769 l 883 5741 l 875 5712 l 867 5683 l 859 5653 l 851 5622 l 844 5591 l 836 5559 l 829 5526 l 821 5493 l 814 5459 l 808 5426 l 801 5391 l 795 5357 l 789 5322 l 784 5288 l 779 5253 l 774 5219 l 770 5185 l 766 5151 l 763 5118 l 760 5085 l 757 5052 l 755 5020 l 753 4988 l 752 4956 l 751 4925 l 750 4894 l 750 4863 l 750 4831 l 751 4800 l 752 4768 l 753 4736 l 755 4704 l 757 4671 l 760 4637 l 763 4603 l 767 4568 l 771 4533 l 776 4498 l 781 4462 l 787 4426 l 793 4390 l 800 4354 l 807 4318 l 814 4282 l 822 4246 l 830 4210 l 839 4175 l 848 4140 l 857 4106 l 866 4072 l 876 4040 l 886 4007 l 896 3976 l 907 3945 l 917 3915 l 928 3885 l 939 3857 l 951 3828 l 963 3800 l 976 3768 l 990 3737 l 1005 3705 l 1020 3674 l 1036 3642 l 1052 3611 l 1069 3579 l 1086 3548 l 1104 3516 l 1122 3485 l 1140 3455 l 1158 3424 l 1177 3394 l 1195 3365 l 1213 3337 l 1231 3310 l 1248 3283 l 1265 3258 l 1282 3234 l 1298 3211 l 1313 3190 l 1328 3169 l 1342 3151 l 1355 3133 l 1367 3117 l 1379 3102 l 1390 3088 l 1400 3075 l 1421 3049 l 1439 3028 l 1455 3010 l 1468 2997 l 1478 2987 l 1486 2980 l 1492 2976 l 1496 2974 l 1498 2973 l 1500 2974 l 1500 2975 l 1500 2974 l 1501 2972 l 1503 2968 l 1506 2961 l 1512 2952 l 1521 2941 l 1533 2927 l 1547 2912 l 1566 2894 l 1588 2875 l 1599 2865 l 1611 2855 l 1625 2844 l 1640 2832 l 1656 2819 l 1672 2806 l 1690 2791 l 1709 2775 l 1729 2759 l 1750 2742 l 1771 2724 l 1793 2705 l 1815 2686 l 1836 2666 l 1858 2646 l 1880 2626 l 1901 2605 l 1922 2585 l 1942 2564 l 1962 2543 l 1981 2522 l cp gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 11925 4650 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 12000 3900 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 12075 3150 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 12000 2400 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 11700 1725 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 11175 1425 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 10650 1425 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 10125 1725 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 9675 2100 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 9300 2475 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 8850 3000 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 8400 3375 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 7950 3750 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 7425 3975 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 6825 4050 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 6375 3750 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 6000 3375 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 5550 2850 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 5175 2400 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 4650 2100 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 4125 1950 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd [60] 0 sd % Ellipse n 3600 1950 525 750 0 360 DrawEllipse gs col0 s gr [] 0 sd % Polyline 30.000 slw n 6000 4800 m 6000 4350 l gs col0 s gr % Polyline gs clippath 10359 4710 m 10719 4800 l 10359 4890 l 10845 4890 l 10845 4710 l cp clip n 6000 4800 m 10800 4800 l gs col0 s gr gr % arrowhead n 10359 4710 m 10719 4800 l 10359 4890 l 10359 4800 l 10359 4710 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 5910 516 m 6000 156 l 6090 516 l 6090 30 l 5910 30 l cp clip n 6000 2325 m 6000 75 l gs col0 s gr gr % arrowhead n 5910 516 m 6000 156 l 6090 516 l 6000 516 l 5910 516 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 3331 6931 m 2989 7076 l 3222 6787 l 2835 7080 l 2943 7224 l cp clip n 6000 4800 m 2925 7125 l gs col0 s gr gr % arrowhead n 3331 6931 m 2989 7076 l 3222 6787 l 3277 6859 l 3331 6931 l cp gs 0.00 setgray ef gr col0 s % Polyline 15.000 slw n 2925 2288 m 2947 2270 l 2969 2252 l 2992 2235 l 3015 2218 l 3039 2202 l 3064 2187 l 3090 2171 l 3117 2156 l 3144 2141 l 3173 2127 l 3202 2114 l 3232 2100 l 3262 2088 l 3294 2075 l 3326 2064 l 3358 2053 l 3391 2042 l 3425 2033 l 3458 2023 l 3492 2015 l 3526 2007 l 3560 2000 l 3595 1994 l 3629 1988 l 3663 1983 l 3697 1978 l 3730 1974 l 3764 1971 l 3798 1968 l 3832 1966 l 3866 1964 l 3900 1963 l 3933 1962 l 3966 1961 l 3999 1961 l 4033 1962 l 4068 1963 l 4104 1965 l 4140 1967 l 4177 1970 l 4214 1973 l 4253 1978 l 4291 1983 l 4331 1988 l 4371 1995 l 4411 2002 l 4451 2010 l 4492 2019 l 4532 2029 l 4573 2040 l 4613 2051 l 4653 2063 l 4692 2076 l 4731 2090 l 4769 2105 l 4807 2120 l 4844 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1852 5023 l 1848 5000 l 1844 4976 l 1841 4952 l 1838 4928 l 1835 4904 l 1832 4881 l 1830 4859 l 1828 4838 l 1827 4818 l 1826 4798 l 1825 4780 l 1825 4763 l 1825 4737 l 1827 4713 l 1829 4690 l 1831 4668 l 1834 4648 l 1838 4630 l 1841 4615 l 1844 4602 l 1846 4594 l 1848 4588 l 1850 4586 l 1850 4588 l 1849 4593 l 1848 4603 l 1845 4616 l 1841 4632 l 1837 4648 l 1834 4665 l 1830 4679 l 1827 4690 l 1826 4697 l 1825 4700 l 1825 4699 l 1826 4695 l 1828 4688 l 1831 4679 l 1835 4667 l 1840 4652 l 1846 4636 l 1853 4618 l 1861 4599 l 1870 4580 l 1879 4560 l 1890 4540 l 1901 4520 l cp gs col0 s gr [] 0 sd $F2psEnd rs %%EndDocument @endspecial 1044 1776 a(Figure)30 b(4.1:)42 b(A)30 b(radial)f (distributions)e(relativ)m(e)j(to)h(a)g(curv)m(e.)299 2237 y Fl(Example)j(4.2.3)46 b Fs(F)-8 b(or)37 b(radial)f (distributions)d(relativ)m(e)k(to)g(a)g(curv)m(e)g(\000,)i(the)e(co)m (v)-5 b(ariance)38 b(tensor)e(is)g(a)299 2350 y(constan)m(t)29 b(m)m(ultiple)c Fm(j)p Fo(\033)1096 2317 y Fn(2)1093 2377 y Fe(?)1152 2350 y Fm(j)j Fs(of)g(the)f(Euclidean)f(scalar)h(pro)s (duct)g(restricted)g(to)h Fo(N)3054 2368 y Fk(F)10 b Fn(\()p Fk(u)p Fn(\))3236 2350 y Fs(\(Example)27 b(2.4.3\).)299 2463 y(Let)33 b Fo(F)43 b Fs(:)30 b Fo(S)681 2430 y Fn(1)750 2463 y Fm(!)f Fl(R)948 2430 y Fk(n)1028 2463 y Fs(b)s(e)k(a)g(constan)m (t)h(sp)s(eed)e(parameterization)h(of)g(\000)g(and)f(set)i Fo(k)e Fs(=)e(2)p Fo(\031)s Fm(j)p Fo(\033)3440 2430 y Fn(2)3437 2490 y Fe(?)3496 2463 y Fm(j)p Fo(=L)3628 2430 y Fn(2)3628 2490 y Fk(F)3688 2463 y Fs(.)48 b(F)-8 b(or)299 2576 y(this)39 b(v)-5 b(alue)39 b(of)h Fo(k)j Fs(w)m(e)e(ha)m(v)m(e)g(that)f(\000)g(is)f(a)h(critical)f(curv)m(e.)69 b(T)-8 b(o)41 b(c)m(hec)m(k)g(this,)h(w)m(e)e(v)m(erify)f(\(4)p Fo(:)p Fs(6\))j(for)e(all)299 2688 y Fo(X)34 b Fm(2)27 b Fl(R)574 2655 y Fk(n)621 2688 y Fs(.)43 b(By)32 b(Example)e(2.4.3,)k (w)m(e)e(ha)m(v)m(e)g Fo(\026)1855 2703 y Fe(?)1941 2688 y Fs(=)26 b(0)32 b(whic)m(h)e(implies)f(\003)2749 2702 y Fn(1)2815 2688 y Fs(=)e(0.)43 b(Using)31 b(De\014nition)f(2.4.2)299 2801 y(to)h(write)f(\003)706 2815 y Fn(2)775 2801 y Fs(in)f(terms)i(of) f Fo(\033)1293 2768 y Fn(2)1290 2829 y Fe(?)1380 2801 y Fs(giv)m(es)645 3093 y Fm(\000)p Fs(\003)779 3107 y Fn(1)818 3093 y Fs(\()p Fo(X)7 b Fs(\))21 b(+)f(\003)1145 3107 y Fn(2)1185 2938 y Fi( )1267 3032 y Fo(d)p Fl(T)p 1267 3072 121 4 v 1282 3156 a Fo(ds)1397 3093 y(;)15 b(X)1519 2938 y Fi(!)1612 3093 y Fm(\000)1713 3032 y Fo(k)s(L)1825 3046 y Fk(F)p 1713 3072 171 4 v 1748 3156 a Fs(2)p Fo(\031)1893 2938 y Fi(*)1976 3032 y Fo(d)p Fl(T)p 1976 3072 121 4 v 1991 3156 a Fo(ds)2106 3093 y(;)g(X)2228 2938 y Fi(+)2328 3093 y Fs(=)25 b(\003)2487 3107 y Fn(0)2526 3093 y Fo(\033)2581 3056 y Fn(2)2578 3116 y Fe(?)2637 2938 y Fi( )2719 3032 y Fo(d)p Fl(T)p 2719 3072 V 2734 3156 a Fo(ds)2849 3093 y(;)15 b(X)2971 2938 y Fi(!)3064 3093 y Fm(\000)3165 3032 y Fo(k)s(L)3277 3046 y Fk(F)p 3165 3072 171 4 v 3200 3156 a Fs(2)p Fo(\031)3346 2938 y Fi(*)3429 3032 y Fo(d)p Fl(T)p 3429 3072 121 4 v 3444 3156 a Fo(ds)3559 3093 y(;)g(X)3681 2938 y Fi(+)3755 3093 y Fo(:)-3481 b Fs(\(4.14\))299 3388 y(Since)35 b Fo(\033)597 3355 y Fn(2)594 3415 y Fe(?)690 3388 y Fs(is)g(a)i(constan) m(t)h(m)m(ultiple)c Fm(j)p Fo(\033)1684 3355 y Fn(2)1681 3415 y Fe(?)1740 3388 y Fm(j)j Fs(of)g(the)f(Euclidean)f(scalar)h(pro)s (duct)g(restricted)g(to)h Fo(N)3714 3406 y Fk(F)10 b Fn(\()p Fk(u)p Fn(\))3868 3388 y Fs(,)299 3500 y(and)30 b(since)f Fo(d)p Fl(T)p Fo(=ds)d Fm(2)f Fo(N)1138 3519 y Fk(F)10 b Fn(\()p Fk(s)p Fn(\))1314 3500 y Fs(w)m(e)31 b(ha)m(v)m(e)1077 3721 y(\(4)p Fo(:)p Fs(14\))c(=)e(\003)1493 3735 y Fn(0)1533 3721 y Fm(j)p Fo(\033)1613 3683 y Fn(2)1610 3744 y Fe(?)1669 3721 y Fm(jh)p Fo(d)p Fl(T)p Fo(=ds;)15 b(X)7 b Fm(i)22 b(\000)e Fs(\()p Fo(k)s(L)2401 3735 y Fk(F)2460 3721 y Fo(=)p Fs(2)p Fo(\031)s Fs(\))p Fm(h)p Fo(d)p Fl(T)p Fo(=ds;)15 b(X)7 b Fm(i)p Fo(:)299 3931 y Fs(W)-8 b(e)24 b(no)m(w)e(compute)h(an)g(explicit)e(form)m(ula)h(for) g(\003)1942 3945 y Fn(0)1982 3931 y Fs(.)38 b(W)-8 b(e)23 b(switc)m(h)g(to)g(normal)e(co)s(ordinates)i(in)e(the)i(in)m(tegral)299 3970 y Fi(R)342 4075 y Fn(\012)412 4043 y Fo(p)p Fs(\()p Fl(y)p Fs(\))p Fo(d)p Fl(y)31 b Fs(and)f(express)g(the)h(result)e(in)g (terms)h(of)h(the)f(transv)m(erse)h(momen)m(ts,)818 4323 y(1)84 b(=)1100 4200 y Fi(Z)1151 4406 y Fn(\012)1221 4323 y Fo(p)p Fs(\()p Fl(y)p Fs(\))p Fo(d)p Fl(y)27 b Fs(=)1617 4200 y Fi(Z)1707 4226 y Fk(L)1755 4237 y Fg(F)1667 4406 y Fn(0)1826 4200 y Fi(Z)1876 4406 y Fn(\012)1927 4420 y Fg(F)8 b Fb(\()p Fg(s)p Fb(\))2074 4323 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(s)p Fs(\))21 b(+)f Fo(v)s Fs(\)\(1)i Fm(\000)d(h)p Fo(v)s(;)c(d)p Fl(T)p Fo(=ds)p Fm(i)p Fs(\))p Fo(dv)21 b(ds)947 4609 y Fs(=)1100 4485 y Fi(Z)1191 4511 y Fk(L)1239 4522 y Fg(F)1151 4691 y Fn(0)1309 4535 y Fi(\000)1351 4609 y Fs(\003)1414 4623 y Fn(0)1474 4609 y Fm(\000)f Fs(\003)1628 4623 y Fn(1)1667 4609 y Fs(\()p Fo(d)p Fl(T)p Fo(=ds)p Fs(\))1992 4535 y Fi(\001)2035 4609 y Fo(ds:)299 4864 y Fs(Since)29 b(\003)599 4878 y Fn(0)669 4864 y Fs(is)h(constan)m(t)h(and)f(\003)1367 4878 y Fn(1)1432 4864 y Fs(=)25 b(0,)31 b(w)m(e)g(ha)m(v)m(e)g(that)g (\003)2232 4878 y Fn(0)2297 4864 y Fs(=)25 b(1)p Fo(=L)2545 4878 y Fk(F)2604 4864 y Fs(.)41 b(Substituting)28 b(in)m(to)i(\(4)p Fo(:)p Fs(14\))j(giv)m(es)987 5074 y(\(4)p Fo(:)p Fs(14\))27 b(=)e(\(1)p Fo(=L)1527 5088 y Fk(F)1587 5074 y Fs(\))p Fm(j)p Fo(\033)1702 5036 y Fn(2)1699 5097 y Fe(?)1759 5074 y Fm(jh)p Fo(d)p Fl(T)p Fo(=ds;)15 b(X)7 b Fm(i)22 b(\000)e Fs(\()p Fo(k)s(L)2491 5088 y Fk(F)2550 5074 y Fo(=)p Fs(2)p Fo(\031)s Fs(\))p Fm(h)p Fo(d)p Fl(T)p Fo(=ds;)15 b(X)7 b Fm(i)p Fo(:)299 5284 y Fs(Since)39 b Fo(k)45 b Fs(=)d(2)p Fo(\031)s Fm(j)p Fo(\033)931 5251 y Fn(2)928 5311 y Fe(?)988 5284 y Fm(j)p Fo(=L)1120 5251 y Fn(2)1120 5310 y Fk(F)1179 5284 y Fs(,)h(w)m(e)e(ha)m(v)m(e)g(that)g (\(4)p Fo(:)p Fs(14\))i(v)-5 b(anishes)39 b(iden)m(tically)-8 b(.)69 b(This)39 b(sho)m(ws)h(that)h(\000)f(is)f(a)299 5396 y(critical)29 b(curv)m(e.)p eop %%Page: 30 38 30 37 bop 0 100 a Fs(30)0 490 y Fl(Example)34 b(4.2.4)46 b Fp(The)35 b(annulus.)50 b Fs(In)32 b(this)g(example)g(w)m(e)i (consider)d(a)j(sp)s(ecial)d(probabilit)m(y)g(distribu-)0 603 y(tion)f(in)f(the)h(plane.)40 b(Let)1211 813 y(\012)25 b(=)g Fm(f)p Fl(y)h Fm(2)f Fl(R)1688 776 y Fn(2)1753 813 y Fs(:)h Fo(r)1845 827 y Fn(1)1909 813 y Fm(\024)f(j)p Fl(y)q Fm(j)g(\024)g Fo(r)2273 827 y Fn(2)2313 813 y Fm(g)p Fo(:)0 1024 y Fs(Assume)30 b(that)h Fo(p)f Fs(dep)s(ends)e(only) i(on)g Fm(j)p Fl(y)q Fm(j)p Fs(.)40 b(Let)1415 1279 y Fo(\027)1466 1241 y Fk(k)1534 1279 y Fs(=)1630 1155 y Fi(Z)1721 1181 y Fk(r)1753 1190 y Fb(2)1681 1361 y Fk(r)1713 1370 y Fb(1)1807 1279 y Fo(\032)1854 1241 y Fk(k)1896 1279 y Fo(p)p Fs(\()p Fo(\032)p Fs(\))p Fo(d\032:)-2178 b Fs(\(4.15\))0 1539 y(W)-8 b(e)32 b(assert)e(that)h(if)f(the)g(spring) f(constan)m(t)i Fo(k)j Fs(satis\014es)1304 1750 y Fo(r)1345 1764 y Fn(1)1409 1750 y Fo(<)25 b Fs(\(1)p Fo(=)p Fs(2)p Fo(\031)g Fm(\000)20 b Fo(\027)1894 1713 y Fn(2)1933 1750 y 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b(giv)m(e)f(an)f(explicit)f(form)m(ula)h(for)g Fo(\027)1411 4926 y Fn(1)1450 4959 y Fs(.)90 b(By)47 b(c)m(hanging)f(to)h(p)s(olar)f (co)s(ordinates)g(in)f(the)i(in)m(tegral)0 4999 y Fi(R)43 5104 y Fn(\012)113 5072 y Fo(p)p Fs(\()p Fl(y)p Fs(\))p Fo(d)p Fl(y)q Fs(,)31 b(w)m(e)g(get)1047 5350 y(1)26 b(=)1214 5227 y Fi(Z)1264 5433 y Fn(\012)1335 5350 y Fo(p)p Fs(\()p Fl(y)p Fs(\))p Fo(d)p Fl(y)g Fs(=)1730 5227 y Fi(Z)1821 5253 y Fn(2)p Fk(\031)1780 5433 y Fn(0)1918 5227 y Fi(Z)2009 5253 y Fk(r)2041 5262 y Fb(2)1969 5433 y Fk(r)2001 5442 y Fb(1)2095 5350 y Fo(p)p Fs(\()p Fo(\032)p Fs(\))p Fo(\032)15 b(d\032)g(d\022)s(;)p eop %%Page: 31 39 31 38 bop 3803 100 a Fs(31)299 490 y(from)30 b(whic)m(h)f(it)h(follo)m (ws)f(that)i(1)26 b(=)f(2)p Fo(\031)s(\027)1678 457 y Fn(1)1718 490 y Fs(,)30 b(or,)h Fo(\027)1961 457 y Fn(1)2025 490 y Fs(=)25 b(1)p Fo(=)p Fs(2)p Fo(\031)s Fs(.)42 b(Substituting)28 b(this)h(in)m(to)i(\(4)p Fo(:)p Fs(18\))h(giv)m(es)1614 702 y Fo(\034)1654 716 y Fk(F)1737 702 y Fs(=)25 b(\()p Fm(\000)p Fo(\027)1990 664 y Fn(2)2050 702 y Fs(+)20 b(1)p Fo(=)p Fs(2)p Fo(\031)s Fs(\))p Fo(=r)g Fl(N)p Fo(:)299 914 y Fs(W)-8 b(e)31 b(no)m(w)g(write)e(\(4)p Fo(:)p Fs(5\))k(m)m(ultiplying)26 b Fo(\034)1619 928 y Fk(F)1708 914 y Fs(b)m(y)1834 836 y Fm(p)p 1910 836 53 4 v 78 x Fo(h=)2007 857 y Fm(p)p 2084 857 47 4 v 2084 914 a Fo(g)j Fs(=)c Fo(r)32 b Fs(and)e(adding)f Fo(k)s Fs(\001)2926 928 y Fk(g)2966 914 y Fo(F)39 b Fs(=)24 b Fm(\000)p Fo(k)s(r)s Fl(N)p Fs(,)1200 1141 y Fo(\034)1240 1155 y Fk(F)1299 1059 y Fm(p)p 1375 1059 53 4 v 82 x Fo(h=)1472 1080 y Fm(p)p 1548 1080 47 4 v 61 x Fo(g)g Fs(+)c Fo(k)s Fs(\001)1832 1155 y Fk(g)1872 1141 y Fo(F)38 b Fs(=)25 b(\()p Fm(\000)p Fo(\027)2221 1103 y Fn(2)2281 1141 y Fs(+)20 b(1)p Fo(=)p Fs(2)p Fo(\031)s Fs(\))p Fl(N)h Fm(\000)f Fo(k)s(r)s Fl(N)p Fo(:)299 1353 y Fs(This)k (expression)h(v)-5 b(anishes)25 b(iden)m(tically)f(since)h Fo(r)j Fs(=)d(\(1)p Fo(=)p Fs(2)p Fo(\031)15 b Fm(\000)d Fo(\027)2478 1320 y Fn(2)2517 1353 y 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5281 y Fm(p)p 1497 5281 53 4 v 82 x Fo(h=)1594 5301 y Fm(p)p 1670 5301 47 4 v 62 x Fo(g)24 b Fs(+)c Fo(k)s Fs(\001)1954 5377 y Fk(g)1994 5363 y Fo(F)38 b Fs(=)25 b(0)p Fo(;)-2256 b Fs(\(4.26\))p eop %%Page: 33 41 33 40 bop 3803 100 a Fs(33)299 490 y(where)31 b(the)h(di\013erence)f (in)g(the)g(v)m(olume)h(elemen)m(t)g(b)s(et)m(w)m(een)g(the)g(t)m(w)m (o)h(in)m(tegrals)f(in)e(\(4)p Fo(:)p Fs(4\))j(is)e(expressed)299 603 y(in)26 b(terms)i(of)g(the)f(quotien)m(t)1263 525 y Fm(p)p 1339 525 53 4 v 78 x Fo(h=)1436 546 y Fm(p)p 1513 546 47 4 v 1513 603 a Fo(g)s Fs(.)40 b(Since)27 b Fo(\034)1899 617 y Fk(F)1985 603 y Fs(is)f(normal)h(to)h Fo(F)13 b Fs(\()p Fo(M)d Fs(\),)30 b(the)e(tangen)m(tial)g(comp)s(onen) m(t)g(of)299 716 y(\001)375 730 y Fk(g)415 716 y Fo(F)43 b Fs(is)30 b(zero,)1852 919 y(\(\001)1963 933 y Fk(g)2003 919 y Fo(F)13 b Fs(\))2109 938 y Fe(k)2174 919 y Fs(=)25 b(0)p Fo(:)-2041 b Fs(\(4.27\))299 1123 y(Therefore,)28 b(a)g(critical)f(em)m(b)s(edding)f Fo(F)40 b Fs(is)27 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(su\016ces)h(to)h(sho)m(w)e(that)i(the)f(map)g Fo(\036)27 b Fs(:)h Fo(S)2657 5251 y Fn(1)2725 5284 y Fm(!)f Fo(S)2904 5251 y Fn(1)2976 5284 y Fs(de\014ned)j(b)m(y)i Fo(\036)c Fs(=)f Fo(\025)3651 5298 y Fk(F)3696 5307 y Fb(0)3756 5284 y Fm(\016)22 b Fo(F)299 5396 y Fs(is)29 b(a)i(di\013eomorphism.)38 b(F)-8 b(or)31 b(this,)f(w)m(e)h(apply)e(the)h(follo)m(wing)f (sublemma.)p eop %%Page: 34 42 34 41 bop 0 100 a Fs(34)0 490 y Fl(Sublemma)33 b(4.2.7)46 b Fp(L)-5 b(et)32 b Fo(F)985 504 y Fn(0)1050 490 y Fm(2)25 b Fo(C)1208 457 y Fe(1)1282 490 y Fs(\()p Fo(S)1378 457 y Fn(1)1418 490 y Fo(;)15 b Ff(R)1536 457 y Fk(n)1583 490 y Fs(\))p Fp(.)42 b(Then)31 b(ther)-5 b(e)32 b(exists)g Fo(\017)25 b(>)g Fs(0)31 b Fp(such)g(that)i(for)e(al)5 b(l)32 b Fo(u)3344 504 y Fn(0)3409 490 y Fm(2)24 b Fo(S)3555 457 y Fn(1)0 603 y Fp(and)35 b(al)5 b(l)35 b Fo(F)41 b Fm(2)27 b Fo(C)569 570 y Fe(1)643 603 y Fs(\()p Fo(S)739 570 y Fn(1)779 603 y Fo(;)15 b Ff(R)898 570 y Fk(n)945 603 y Fs(\))34 b Fp(with)h Fm(jj)p Fo(F)g Fm(\000)21 b Fo(F)1506 617 y Fn(0)1546 603 y Fm(jj)1596 622 y Fk(C)1651 604 y Fb(1)1718 603 y Fo(<)28 b(\017)p Fp(,)34 b(the)g(angle)h(b)-5 b(etwe)g(en)34 b Fl(T)2713 617 y Fn(0)2753 603 y Fs(\()p Fo(u)2840 617 y Fn(0)2879 603 y Fs(\))h Fp(and)g Fl(T)p Fs(\()p Fo(u)p Fs(\))f Fp(is)g(less)0 716 y(that)g Fo(\031)s(=)p Fs(2)p Fp(,)f(for)g(al)5 b(l)34 b Fo(u)25 b Fm(2)g Fo(\036)885 683 y Fe(\000)p Fn(1)979 716 y Fm(f)p Fo(u)1076 730 y Fn(0)1116 716 y Fm(g)p Fp(.)0 920 y(Pr)-5 b(o)g(of)34 b(of)f(Sublemma)g(4.2.7)p Fs(.)136 1128 y(F)-8 b(or)31 b Fo(F)44 b Fs(in)29 b(a)i Fo(C)655 1095 y Fn(1)724 1128 y Fs(neigh)m(b)s(orho)s(o)s(d)d(of)i Fo(F)1456 1142 y Fn(0)1527 1128 y Fs(and)f Fo(u)d Fm(2)f Fo(\036)1921 1095 y Fe(\000)p Fn(1)2015 1128 y Fm(f)p Fo(u)2112 1142 y Fn(0)2152 1128 y Fm(g)p Fs(,)31 b(w)m(e)g(sho)m(w)f(that)1348 1324 y Fm(j)p Fl(T)1446 1338 y Fn(0)1485 1324 y Fs(\()p Fo(u)1572 1338 y Fn(0)1612 1324 y Fs(\))21 b Fm(\000)f Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)26 b Fo(<)2101 1245 y Fm(p)p 2176 1245 46 4 v 2176 1324 a Fs(2)q Fo(:)-2247 b Fs(\(4.30\))0 1532 y(Let)26 b Fo(\017)195 1546 y Fn(1)259 1532 y Fo(>)f Fs(0)h(suc)m(h)f(that)g(if)f Fm(jj)p Fo(F)g Fm(\000)10 b Fo(F)1166 1546 y Fn(0)1205 1532 y Fm(jj)1255 1552 y Fk(C)1310 1533 y Fb(1)1375 1532 y Fo(<)25 b(\017)1508 1546 y Fn(1)1572 1532 y Fs(then)g Fm(jj)p Fl(T)10 b Fm(\000)g Fl(T)2061 1546 y Fn(0)2099 1532 y Fm(jj)2149 1552 y Fk(C)2204 1533 y Fb(0)2269 1532 y Fo(<)25 b Fs(1)p Fo(=)2455 1457 y Fm(p)p 2532 1457 V 2532 1532 a Fs(2.)39 b(Since)24 b Fl(T)2946 1546 y Fn(0)2995 1532 y Fm(\016)10 b Fo(\025)3103 1546 y Fk(F)3148 1555 y Fb(0)3212 1532 y Fs(:)25 b(\012)g Fm(!)g Fl(R)3548 1499 y Fk(n)0 1645 y Fs(is)32 b(a)h(uniformly)c(con)m (tin)m(uous)j(function)g(on)g(the)h(compact)h(\012,)f(there)f(exists)g Fo(\017)2705 1659 y Fn(2)2774 1645 y Fo(>)c Fs(0)33 b(suc)m(h)f(that)i (for)e(all)0 1758 y Fl(y)55 1772 y Fn(1)95 1758 y Fo(;)15 b Fl(y)190 1772 y Fn(2)266 1758 y Fm(2)36 b Fs(\012)h(with)f Fm(j)p Fl(y)761 1772 y Fn(1)825 1758 y Fm(\000)24 b Fl(y)975 1772 y Fn(2)1015 1758 y Fm(j)37 b Fo(<)f(\017)1221 1772 y Fn(2)1260 1758 y Fs(,)j(w)m(e)f(ha)m(v)m(e)g Fm(j)p Fl(T)1779 1772 y Fn(0)1844 1758 y Fm(\016)25 b Fo(\025)1967 1772 y Fk(F)2012 1781 y Fb(0)2050 1758 y Fs(\()p Fl(y)2141 1772 y Fn(1)2180 1758 y Fs(\))g Fm(\000)g Fl(T)2409 1772 y Fn(0)2473 1758 y Fm(\016)g Fo(\025)2596 1772 y Fk(F)2641 1781 y Fb(0)2679 1758 y Fs(\()p Fl(y)2770 1772 y Fn(2)2809 1758 y Fs(\))p Fm(j)37 b Fo(<)g Fs(1)p Fo(=)3104 1683 y Fm(p)p 3180 1683 V 75 x Fs(2)q(.)61 b(Setting)0 1871 y Fo(\017)25 b Fs(=)g(min)o Fm(f)p Fo(\017)392 1885 y Fn(1)431 1871 y Fo(;)15 b(\017)508 1885 y Fn(2)548 1871 y Fm(g)p Fs(,)31 b(w)m(e)g(ha)m(v)m(e)230 2068 y Fm(j)p Fl(T)328 2082 y Fn(0)368 2068 y Fs(\()p Fo(u)455 2082 y Fn(0)494 2068 y Fs(\))21 b Fm(\000)f Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)84 b Fs(=)e Fm(j)p Fl(T)1196 2082 y Fn(0)1236 2068 y Fs(\()p Fo(\036)p Fs(\()p Fo(u)p Fs(\)\))22 b Fm(\000)d Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)945 2206 y(\024)82 b(j)p Fl(T)1196 2220 y Fn(0)1236 2206 y Fs(\()p Fo(\036)p Fs(\()p Fo(u)p Fs(\)\))22 b Fm(\000)d Fl(T)1667 2220 y Fn(0)1707 2206 y Fs(\()p Fo(\025)1795 2220 y Fk(F)1840 2229 y Fb(0)1899 2206 y Fm(\016)h Fo(F)2022 2220 y Fn(0)2062 2206 y Fs(\()p Fo(u)p Fs(\)\))p Fm(j)i Fs(+)e Fm(j)p Fl(T)2455 2220 y Fn(0)2494 2206 y Fs(\()p Fo(\025)2582 2220 y Fk(F)2627 2229 y Fb(0)2686 2206 y Fm(\016)h Fo(F)2810 2220 y Fn(0)2850 2206 y Fs(\()p Fo(u)p Fs(\)\))g Fm(\000)f Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)p Fo(:)0 2402 y Fs(W)-8 b(riting)30 b Fo(\036)25 b Fs(=)g Fo(\025)562 2416 y Fk(F)607 2425 y Fb(0)666 2402 y Fm(\016)20 b Fo(F)13 b Fs(,)31 b(and)f(since)f Fm(j)p Fo(F)13 b Fs(\()p Fo(u)p Fs(\))22 b Fm(\000)e Fo(F)1646 2416 y Fn(0)1685 2402 y Fs(\()p Fo(u)p Fs(\))p Fm(j)27 b Fo(<)e(\017)g Fm(\024)g Fo(\017)2150 2416 y Fn(2)2189 2402 y Fs(,)31 b(w)m(e)g(get)755 2598 y Fm(j)p Fl(T)853 2612 y Fn(0)893 2598 y Fs(\()p Fo(u)980 2612 y Fn(0)1020 2598 y Fs(\))20 b Fm(\000)g Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)26 b Fo(<)f Fs(1)p Fo(=)1598 2519 y Fm(p)p 1675 2519 V 1675 2598 a Fs(2)20 b(+)g Fm(j)p Fl(T)1929 2612 y Fn(0)1969 2598 y Fs(\()p Fo(\025)2057 2612 y Fk(F)2102 2621 y Fb(0)2161 2598 y Fm(\016)g Fo(F)2284 2612 y Fn(0)2324 2598 y Fs(\()p Fo(u)p Fs(\)\))i Fm(\000)d Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)p Fo(:)0 2795 y Fs(Since)29 b Fo(\025)290 2809 y Fk(F)335 2818 y Fb(0)394 2795 y Fm(\016)21 b Fo(F)518 2809 y Fn(0)583 2795 y Fs(:)k Fo(S)694 2762 y Fn(1)759 2795 y Fm(!)g Fo(S)936 2762 y Fn(1)1006 2795 y Fs(is)k(the)i(iden)m(tit)m(y)f(map)951 2991 y Fm(j)p Fl(T)1049 3005 y Fn(0)1088 2991 y Fs(\()p Fo(u)1175 3005 y Fn(0)1215 2991 y Fs(\))21 b Fm(\000)f Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)26 b Fo(<)f Fs(1)p Fo(=)1794 2912 y Fm(p)p 1870 2912 V 79 x Fs(2)c(+)f Fm(j)p Fl(T)2125 3005 y Fn(0)2164 2991 y Fs(\()p Fo(u)p Fs(\))h Fm(\000)f Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)p Fo(:)0 3188 y Fs(Using)30 b Fm(j)p Fl(T)354 3202 y Fn(0)393 3188 y Fs(\()p Fo(u)p Fs(\))21 b Fm(\000)f Fl(T)p Fs(\()p Fo(u)p Fs(\))p Fm(j)26 b Fo(<)f Fs(1)p Fo(=)1059 3113 y Fm(p)p 1135 3113 V 75 x Fs(2)q(,)30 b(giv)m(es)h(\(4)p Fo(:)p Fs(30\).)3280 3384 y Fl(Q.E.D.)0 3581 y Fs(W)-8 b(e)33 b(no)m(w)e(con)m(tin)m(ue)h(with)e(the)i(pro)s(of)f(of)g(Lemma)h (4.2.6.)46 b(W)-8 b(e)33 b(use)e(Sublemma)f(4.2.7)j(to)f(pro)m(v)m(e)h (that)f Fo(\036)0 3694 y Fs(is)d(a)i(di\013eomorphism.)136 3807 y(W)-8 b(e)23 b(start)e(b)m(y)g(sho)m(wing)f(that)i Fo(\036)f Fs(is)f(injectiv)m(e.)37 b(Let)22 b Fo(u)1890 3821 y Fn(0)1955 3807 y Fm(2)j Fo(S)2102 3774 y Fn(1)2162 3807 y Fs(and)20 b(consider)g(the)i(subset)e Fm(f)p Fo(u)3188 3821 y Fk(i)3217 3807 y Fm(g)3262 3821 y Fk(i)p Fe(2)p Fk(I)3398 3807 y Fm(\032)25 b Fo(S)3555 3774 y Fn(1)0 3920 y Fs(suc)m(h)37 b(that)g Fo(\036)p Fs(\()p Fo(u)556 3934 y Fk(i)585 3920 y Fs(\))f(=)g Fo(u)815 3934 y Fn(0)855 3920 y Fs(.)60 b(Then)36 b Fo(u)1236 3934 y Fk(i)1301 3920 y Fm(2)f Fo(\036)1451 3887 y Fe(\000)p Fn(1)1546 3920 y Fm(f)p Fo(u)1643 3934 y Fn(0)1683 3920 y Fm(g)i Fs(and)g Fo(F)13 b Fs(\()p Fo(u)2107 3934 y Fk(i)2135 3920 y Fs(\))37 b Fm(2)f Fo(\025)2357 3881 y Fe(\000)p Fn(1)2357 3948 y Fk(F)2402 3957 y Fb(0)2451 3920 y Fm(f)p Fo(u)2548 3934 y Fn(0)2588 3920 y Fm(g)h Fs(for)g(all)f Fo(i)g Fm(2)g Fo(I)7 b Fs(.)61 b(Consider)0 4042 y(no)m(w)33 b(the)f(plane)g Fo(N)668 4060 y Fk(F)713 4069 y Fb(0)747 4060 y Fn(\()p Fk(u)815 4069 y Fb(0)850 4060 y Fn(\))914 4042 y Fs(orthogonal)h(to)g Fo(F)1542 4056 y Fn(0)1582 4042 y Fs(\()p Fo(S)1678 4009 y Fn(1)1718 4042 y Fs(\))g(at)g Fo(F)1957 4056 y Fn(0)1997 4042 y Fs(\()p Fo(u)2084 4056 y Fn(0)2124 4042 y Fs(\).)47 b(Then)32 b Fm(f)p Fo(F)13 b Fs(\()p Fo(u)2674 4056 y Fk(i)2703 4042 y Fs(\))p Fm(g)29 b(\032)g Fo(N)2985 4060 y Fk(F)3030 4069 y Fb(0)3064 4060 y Fn(\()p Fk(u)3132 4069 y Fb(0)3167 4060 y Fn(\))3231 4042 y Fs(is)j(the)g(set)0 4154 y(of)38 b(p)s(oin)m(ts)f(at)i(whic)m(h) e Fo(F)13 b Fs(\()p Fo(S)946 4121 y Fn(1)986 4154 y Fs(\))38 b(in)m(tersects)h Fo(N)1543 4173 y Fk(F)1588 4182 y Fb(0)1622 4173 y Fn(\()p Fk(u)1690 4182 y Fb(0)1725 4173 y Fn(\))1757 4154 y Fs(.)64 b(By)38 b(Sublemma)e(4.2.7,)42 b(the)d(angle)f(b)s(et)m (w)m(een)h(the)0 4267 y(tangen)m(t)c(v)m(ectors)h Fl(T)720 4281 y Fn(0)759 4267 y Fs(\()p Fo(u)846 4281 y Fn(0)886 4267 y Fs(\))e(and)g Fl(T)p Fs(\()p Fo(u)1296 4281 y Fk(i)1324 4267 y Fs(\))g(is)f(less)h(than)g Fo(\031)s(=)p Fs(2,)i(for)d(all)g Fo(i)f Fm(2)f Fo(I)7 b Fs(.)52 b(This)32 b(can)i(only)f(happ)s(en)g(if)0 4380 y Fm(f)p Fo(F)13 b Fs(\()p Fo(u)203 4394 y Fk(i)232 4380 y Fs(\))p Fm(g)31 b Fs(consists)f(of)h(one)f(elemen)m(t)h(only)-8 b(.)40 b(This)29 b(sho)m(ws)h(that)h Fo(\036)g Fs(is)e(injectiv)m(e.)136 4493 y(Next)41 b(w)m(e)f(sho)m(w)f(that)h Fo(\036)f Fs(is)g(surjectiv)m (e.)68 b(Since)38 b Fo(\036)j Fs(=)f Fo(\025)2120 4507 y Fk(F)2165 4516 y Fb(0)2230 4493 y Fm(\016)26 b Fo(F)53 b Fs(is)38 b(con)m(tin)m(uous)h(and)g(since)g Fo(S)3455 4460 y Fn(1)3534 4493 y Fs(is)0 4606 y(Hausdor\013)j(and)g(compact,)48 b(b)m(y)42 b(the)h(Closed)f(Map)h(Lemma)g Fo(\036)p Fs(\()p Fo(S)2343 4573 y Fn(1)2383 4606 y Fs(\))g(is)e(closed.)78 b(In)42 b(particular,)i(w)m(e)0 4719 y(ha)m(v)m(e)d(that)g Fo(\036)p Fs(\()p Fo(S)576 4686 y Fn(1)616 4719 y Fs(\))f(con)m(tains)g (its)f(b)s(oundary)f Fo(@)5 b(\036)p Fs(\()p Fo(S)1815 4686 y Fn(1)1855 4719 y Fs(\))42 b Fm(\032)e Fo(S)2104 4686 y Fn(1)2144 4719 y Fs(.)69 b(Assume)39 b(that)i Fo(@)5 b(\036)p Fs(\()p Fo(S)2994 4686 y Fn(1)3034 4719 y Fs(\))42 b Fm(6)p Fs(=)e Fm(;)h Fs(and)e(let)0 4832 y Fo(u)52 4846 y Fn(1)129 4832 y Fm(2)d Fo(S)287 4799 y Fn(1)364 4832 y Fs(with)g Fo(\036)p Fs(\()p Fo(u)719 4846 y Fn(1)759 4832 y Fs(\))h Fm(2)g Fo(@)5 b(\036)p Fs(\()p Fo(S)1132 4799 y Fn(1)1172 4832 y Fs(\).)63 b(Set)37 b Fo(u)1510 4846 y Fn(0)1587 4832 y Fs(=)f Fo(\036)p Fs(\()p Fo(u)1835 4846 y Fn(1)1875 4832 y Fs(\).)63 b(Then)36 b Fo(u)2294 4846 y Fn(1)2371 4832 y Fm(2)g Fo(\036)2522 4799 y Fe(\000)p Fn(1)2617 4832 y Fm(f)p Fo(u)2714 4846 y Fn(0)2754 4832 y Fm(g)p Fs(,)k(and)c(b)m(y)i(Sublemma)0 4945 y(4.2.7,)32 b(the)e(angle)g(b)s(et)m(w)m(een)h Fl(T)1059 4959 y Fn(0)1098 4945 y Fs(\()p Fo(u)1185 4959 y Fn(0)1225 4945 y Fs(\))f(and)g Fl(T)p Fs(\()p Fo(u)1627 4959 y Fn(1)1666 4945 y Fs(\))h(is)e(less)g(that)i Fo(\031)s(=)p Fs(2.)41 b(In)30 b(particular,)e(there)i(is)g(an)f(op)s(en)0 5058 y(segmen)m(t)e(near)g Fo(F)13 b Fs(\()p Fo(u)704 5072 y Fn(1)744 5058 y Fs(\))26 b(in)f Fo(F)13 b Fs(\()p Fo(S)1074 5025 y Fn(1)1114 5058 y Fs(\))26 b(that)h(pro)5 b(jects)27 b(on)m(to)g(an)g(op)s(en)e(segmen)m(t)j(of)e Fo(F)2754 5072 y Fn(0)2794 5058 y Fs(\()p Fo(u)2881 5072 y Fn(0)2921 5058 y Fs(\))g(in)f Fo(F)3142 5072 y Fn(0)3182 5058 y Fs(\()p Fo(S)3278 5025 y Fn(1)3318 5058 y Fs(\).)40 b(This)0 5171 y(con)m(tradicts)31 b(that)g Fo(\036)p Fs(\()p Fo(u)806 5185 y Fn(1)846 5171 y Fs(\))26 b Fm(2)e Fo(@)5 b(\036)p Fs(\()p Fo(S)1195 5138 y Fn(1)1236 5171 y Fs(\),)31 b(and)e(then)h Fo(@)5 b(\036)p Fs(\()p Fo(S)1913 5138 y Fn(1)1954 5171 y Fs(\))25 b(=)g Fm(;)p Fs(.)41 b(This)29 b(sho)m(ws)h(that)h Fo(\036)f Fs(is)g(surjectiv)m(e.)136 5284 y(F)-8 b(rom)22 b(the)g(Homeomorphism)e(Theorem,)j(it)e(follo)m (ws)g(that)h(the)f(con)m(tin)m(uous)g(bijection)g Fo(\036)k Fs(:)h Fo(S)3314 5251 y Fn(1)3378 5284 y Fm(!)f Fo(S)3555 5251 y Fn(1)0 5396 y Fs(from)39 b(the)g(compact)i(and)e(Hausdor\013)g (space)h Fo(S)1685 5363 y Fn(1)1763 5396 y Fs(on)m(to)h(itself)d(is)g (a)i(homeomorphism.)66 b(A)39 b(su\016cien)m(t)p eop %%Page: 35 43 35 42 bop 3803 100 a Fs(35)299 490 y(condition)34 b(for)h(a)g(smo)s (oth)h(homeomorphism)d(to)j(b)s(e)f(a)g(di\013eomorphism)e(is)h(that)i (its)f(di\013eren)m(tial)e(b)s(e)299 603 y(nonsingular)d(at)k(ev)m(ery) f(p)s(oin)m(t)f(of)h Fo(S)1555 570 y Fn(1)1595 603 y Fs(.)48 b(Since)32 b Fo(F)46 b Fs(is)32 b(an)h(em)m(b)s(edding,)f(to)h (sho)m(w)g(that)h Fo(\036)3318 617 y Fe(\003)3387 603 y Fs(=)29 b(\()p Fo(\025)3575 617 y Fk(F)3620 626 y Fb(0)3681 603 y Fm(\016)22 b Fo(F)13 b Fs(\))3854 617 y Fe(\003)299 716 y Fs(is)43 b(injectiv)m(e,)k(it)c(su\016ces)g(to)i(sho)m(w)e(that)i Fo(\025)1860 730 y Fk(F)1905 739 y Fb(0)1943 743 y Fe(\003)1983 716 y Fl(T)p Fs(\()p Fo(u)p Fs(\))j Fm(6)p Fs(=)f(0,)h(for)43 b(all)g Fo(u)k Fm(2)g Fo(S)3022 683 y Fn(1)3062 716 y Fs(.)81 b(If)43 b Fo(\025)3325 730 y Fk(F)3370 739 y Fb(0)3408 743 y Fe(\003)3448 716 y Fl(T)p Fs(\()p Fo(u)3608 730 y Fn(1)3647 716 y Fs(\))48 b(=)f(0)299 829 y(for)42 b(some)i Fo(u)743 843 y Fn(1)828 829 y Fm(2)i Fo(S)996 796 y Fn(1)1035 829 y Fs(,)g(the)d(angle)g(b)s(et)m(w)m(een)h Fl(T)p Fs(\()p Fo(u)2049 843 y Fn(1)2088 829 y Fs(\))g(and)e Fl(T)2429 843 y Fn(0)2468 829 y Fs(\()p Fo(\025)2556 843 y Fk(F)2601 852 y Fb(0)2668 829 y Fm(\016)29 b Fo(F)13 b Fs(\()p Fo(u)2900 843 y Fn(1)2940 829 y Fs(\)\))44 b(w)m(ould)d(b)s(e)h Fo(\031)s(=)p Fs(2.)79 b(This)299 941 y(con)m(tradicts)32 b(Sublemma)e(4.2.7)j(and)e(therefore)h Fo(\025)2046 955 y Fk(F)2091 964 y Fb(0)2129 969 y Fe(\003)2169 941 y Fl(T)p Fs(\()p Fo(u)p Fs(\))27 b Fm(6)p Fs(=)g(0)32 b(for)f(all)g Fo(u)c Fm(2)g Fo(S)3062 908 y Fn(1)3101 941 y Fs(.)44 b(This)30 b(sho)m(ws)h(that)i Fo(\036)299 1054 y Fs(is)c(a)i(di\013eomorphism.)435 1167 y(W)-8 b(e)28 b(no)m(w)e(pro)m(v)m(e)h(the)g(second)f(statemen)m(t)i(of)f(the) f(lemma;)i(i.e.,)f(that)g Fo(F)40 b Fs(has)26 b(constan)m(t)h(sp)s(eed) f(if)f(and)299 1280 y(only)k(if)h Fo(\036)636 1247 y Fe(\000)p Fn(1)761 1280 y Fs(is)f(giv)m(en)h(b)m(y)h(\(4)p Fo(:)p Fs(29\).)435 1393 y(View)36 b Fo(\036)h Fs(as)g(a)g(real-v)-5 b(alued)36 b(function)g(de\014ned)f(on)i(the)g(in)m(terv)-5 b(al)36 b([0)p Fo(;)15 b Fs(2)p Fo(\031)s Fs(].)61 b(Without)37 b(loss)f(of)h(gen-)299 1506 y(eralit)m(y)-8 b(,)41 b(let)e Fo(\036)p Fs(\(0\))h(=)f(0.)66 b(W)-8 b(e)40 b(compute)f Fm(j)p Fo(F)1843 1473 y Fe(0)1867 1506 y Fm(j)f Fs(b)m(y)h(applying)e (the)h(c)m(hain)h(rule)e(in)g(\(4)p Fo(:)p Fs(28\).)68 b(This)37 b(giv)m(es)299 1619 y Fm(j)p Fo(F)395 1586 y Fe(0)419 1619 y Fm(j)25 b Fs(=)g Fm(j)p Fs(\()p Fo(F)696 1586 y Fe(0)683 1643 y Fn(0)744 1619 y Fs(+)20 b Fo(V)908 1586 y Fe(0)931 1619 y Fs(\))h Fm(\016)f Fo(\036)p Fm(jj)p Fo(\036)1210 1586 y Fe(0)1234 1619 y Fm(j)p Fs(.)41 b(Since)30 b Fo(\036)g Fs(is)g(a)g(di\013eomorphism)e(and)i Fo(\036)p Fs(\(0\))c(=)f(0,)31 b(then)f Fo(\036)3284 1586 y Fe(0)3333 1619 y Fo(>)25 b Fs(0.)41 b(Hence,)1629 1820 y Fm(j)p Fo(F)1725 1782 y Fe(0)1748 1820 y Fm(j)26 b Fs(=)f Fm(j)p Fs(\()p Fo(F)2026 1782 y Fe(0)2013 1842 y Fn(0)2073 1820 y Fs(+)20 b Fo(V)2237 1782 y Fe(0)2261 1820 y Fs(\))g Fm(\016)h Fo(\036)p Fm(j)p Fo(\036)2515 1782 y Fe(0)2539 1820 y Fo(:)-2265 b Fs(\(4.31\))299 2021 y(W)-8 b(e)34 b(use)f(the)g(form)m(ula)g(for)g(the)g(deriv)-5 b(ativ)m(e)33 b(of)g(the)h(in)m(v)m(erse)f(function)f(\()p Fo(\036)2853 1988 y Fe(\000)p Fn(1)2947 2021 y Fs(\))2982 1988 y Fe(0)3028 2021 y Fm(\016)23 b Fo(\036)30 b Fs(=)f(1)p Fo(=\036)3424 1988 y Fe(0)3482 2021 y Fs(and)j(\(4)p Fo(:)p Fs(31\))299 2134 y(to)d(obtain)e(\()p Fo(\036)777 2101 y Fe(\000)p Fn(1)872 2134 y Fs(\))907 2101 y Fe(0)947 2134 y Fm(\016)16 b Fo(\036)25 b Fs(=)g(\(1)p Fo(=)p Fm(j)p Fo(F)1404 2101 y Fe(0)1429 2134 y Fm(j)p Fs(\))p Fm(j)p Fo(F)1585 2101 y Fe(0)1572 2158 y Fn(0)1629 2134 y Fs(+)15 b Fo(V)1789 2101 y Fe(0)1812 2134 y Fm(j)h(\016)g Fo(\036)p Fs(.)40 b(Comp)s(osing)27 b(with)g Fo(\036)h Fs(on)g(the)g(righ)m(t)g(and)g(in) m(tegrating)299 2247 y(from)i(0)h(to)g Fo(u)25 b Fm(2)g Fs([0)p Fo(;)15 b Fs(2)p Fo(\031)s Fs(])32 b(giv)m(es)1119 2492 y Fo(\036)1173 2454 y Fe(\000)p Fn(1)1267 2492 y Fs(\()p Fo(u)p Fs(\))21 b Fm(\000)f Fo(\036)1555 2454 y Fe(\000)p Fn(1)1650 2492 y Fs(\(0\))26 b(=)1887 2368 y Fi(Z)1978 2394 y Fk(u)1938 2574 y Fn(0)2038 2492 y Fm(j)p Fo(F)2134 2454 y Fe(0)2121 2514 y Fn(0)2161 2492 y Fs(\()p Fo(v)s Fs(\))c(+)d Fo(V)2464 2454 y Fe(0)2487 2492 y Fs(\()p Fo(v)s Fs(\))p Fm(j)p Fo(=)p Fm(j)p Fo(F)2770 2454 y Fe(0)2795 2492 y Fs(\()p Fo(v)s Fs(\))p Fm(j)c Fo(dv)s(;)299 2740 y Fs(since)36 b Fo(\036)p Fs(\(0\))i(=)e(0.)62 b(If)36 b Fm(j)p Fo(F)1167 2707 y Fe(0)1191 2740 y Fm(j)h Fs(is)f(constan)m(t,)41 b(then)c(this)f(form)m(ula)g(implies)e(\(4)p Fo(:)p Fs(29\).)63 b(Con)m(v)m(ersely)-8 b(,)40 b(assume)299 2853 y(that)31 b(\(4)p Fo(:)p Fs(29\))h(holds.)40 b(Then)29 b(w)m(e)i(ha)m(v)m(e)872 2975 y Fi(Z)963 3001 y Fk(u)923 3181 y Fn(0)1023 3098 y Fm(j)p Fo(F)1119 3061 y Fe(0)1106 3121 y Fn(0)1146 3098 y Fs(\()p Fo(v)s Fs(\))22 b(+)e Fo(V)1449 3061 y Fe(0)1472 3098 y Fs(\()p Fo(v)s Fs(\))p Fm(j)p Fo(=)p Fm(j)p Fo(F)1755 3061 y Fe(0)1780 3098 y Fs(\()p Fo(v)s Fs(\))p Fm(j)15 b Fo(dv)31 b Fs(=)2276 3037 y(1)p 2165 3077 268 4 v 2165 3161 a Fm(j)p Fo(F)2261 3134 y Fe(0)2284 3161 y Fs(\()p Fo(u)p Fs(\))p Fm(j)2457 2975 y Fi(Z)2548 3001 y Fk(u)2508 3181 y Fn(0)2608 3098 y Fm(j)p Fo(F)2704 3061 y Fe(0)2691 3121 y Fn(0)2732 3098 y Fs(\()p Fo(v)s Fs(\))21 b(+)f Fo(V)3034 3061 y Fe(0)3057 3098 y Fs(\()p Fo(v)s Fs(\))p Fm(j)p Fo(dv)s(:)299 3350 y Fs(By)34 b(taking)h(the)f(deriv)-5 b(ativ)m(e)34 b(with)f(resp)s(ect)h(to)h Fo(u)f Fs(and)g(simplifying,)d(w)m(e)j(get)i Fm(j)p Fo(F)3103 3317 y Fe(0)3126 3350 y Fm(j)3151 3317 y Fe(0)3207 3350 y Fs(=)31 b(0.)53 b(This)32 b(pro)m(v)m(es)299 3463 y(the)e(lemma.)299 3712 y Fc(4.3)105 b(Second)36 b(v)-6 b(ariation)35 b(of)g(E)299 3915 y Fs(W)-8 b(e)30 b(no)m(w)f(in)m(v)m(estigate)h(conditions)d(under)g(whic)m(h)h(a)h (critical)e(em)m(b)s(edding)g(yields)g(a)i(lo)s(cal)g(minim)m(um)c(of) 299 4028 y Fo(E)36 b Fs(b)m(y)30 b(computing)f(its)h(second)g(v)-5 b(ariation.)435 4141 y(Denote)30 b(b)m(y)e Fo(g)915 4108 y Fe(\003)984 4141 y Fs(the)g(induced)e(metric)i(in)f(the)i(cotangen)m (t)i(bundle)26 b Fo(T)2791 4108 y Fe(\003)2778 4163 y Fk(x)2830 4141 y Fo(M)10 b Fs(.)40 b(Our)27 b(form)m(ulation)g(of)h (the)299 4254 y(second)i(v)-5 b(ariation)30 b(of)g Fo(E)36 b Fs(in)m(v)m(olv)m(es)30 b(t)m(w)m(o)i(additional)c(quan)m(tities.)385 4463 y(\(i\))45 b(The)30 b(symmetric)g(functional)e Fo(Q)1654 4477 y Fk(F)1738 4463 y Fs(:)e Fo(C)1861 4430 y Fe(1)1935 4463 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2182 4430 y Fk(n)2229 4463 y Fs(\))21 b Fm(\002)f Fo(C)2448 4430 y Fe(1)2522 4463 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2769 4430 y Fk(n)2816 4463 y Fs(\))26 b Fm(!)f Fo(C)3065 4430 y Fe(1)3139 4463 y Fs(\()p Fo(M)10 b Fs(\))31 b(giv)m(en)f(b)m(y)1087 4664 y Fo(Q)1159 4678 y Fk(F)1217 4664 y Fs(\()p Fo(X)r(;)15 b(Y)22 b Fs(\))j(=)g Fo(k)s(\016)1693 4626 y Fk(ij)1755 4664 y Fm(h)p Fo(dX)7 b Fs(\()p Fo(e)1996 4626 y Fk(M)1996 4686 y(i)2076 4664 y Fs(\))p Fo(;)15 b(d)5 b(Y)22 b Fs(\()p Fo(e)2355 4626 y Fk(M)2355 4686 y(j)2434 4664 y Fs(\))p Fm(i)f(\000)f Fo(H)2692 4678 y Fk(F)2750 4664 y Fs(\()p Fo(X)r(;)15 b(Y)21 b Fs(\))3011 4582 y Fm(p)p 3088 4582 53 4 v 3088 4664 a Fo(h=)3185 4603 y Fm(p)p 3261 4603 47 4 v 61 x Fo(g)t(:)-2807 b Fs(\(4.32\))359 4922 y(\(ii\))45 b(The)30 b(bilinear)e(map)i Fo(q)1285 4936 y Fk(F)1368 4922 y Fs(:)c(\()p Fl(R)1532 4889 y Fk(n)1599 4922 y Fm(\012)20 b Fo(T)1756 4889 y Fe(\003)1796 4922 y Fo(M)10 b Fs(\))20 b Fm(\002)g Fs(\()p Fl(R)2154 4889 y Fk(n)2221 4922 y Fm(\012)g Fo(T)2378 4889 y Fe(\003)2417 4922 y Fo(M)10 b Fs(\))26 b Fm(!)f Fo(C)2764 4889 y Fe(1)2838 4922 y Fs(\()p Fo(M)10 b Fs(\))31 b(giv)m(en)g(b)m(y)866 5123 y Fo(q)907 5137 y Fk(F)965 5123 y Fs(\()p Fo(X)d Fm(\012)20 b Fo(\030)t(;)15 b(Y)40 b Fm(\012)20 b Fo(\021)s Fs(\))26 b(=)f Fo(k)s Fm(h)p Fo(X)r(;)15 b(Y)22 b Fm(i)p Fo(g)2025 5086 y Fe(\003)2065 5123 y Fs(\()p Fo(\030)t(;)15 b(\021)s Fs(\))1048 5303 y Fm(\000)1134 5179 y Fi(Z)1184 5385 y Fn(\012)1235 5396 y Fg(F)1290 5303 y Fm(h)p Fo(X)r(;)g(v)s Fm(ih)p Fo(Y)5 b(;)15 b(v)s Fm(ih)p Fo(\030)1818 5265 y Fn(#)1884 5303 y Fo(;)g Fs(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2174 5317 y Fk(v)2215 5303 y Fs(\))2250 5265 y Fe(\000)p Fn(1)2345 5303 y Fo(\021)2393 5265 y Fn(#)2456 5303 y Fm(i)p Fo(p)p Fs(\()p Fo(F)34 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)27 b Fm(\000)20 b Fo(S)3228 5317 y Fk(v)3269 5303 y Fs(\))p Fo(dv)3398 5221 y Fm(p)p 3475 5221 53 4 v 3475 5303 a Fo(h=)3572 5242 y Fm(p)p 3648 5242 47 4 v 61 x Fo(g)t(:)-3194 b Fs(\(4.33\))p eop %%Page: 36 44 36 43 bop 0 100 a Fs(36)0 490 y(Our)29 b(main)g(goal)i(in)e(this)h (section)g(is)g(to)h(pro)m(v)m(e)g(the)f(follo)m(wing:)0 651 y Fl(Theorem)k(4.3.1)46 b Fp(L)-5 b(et)32 b Fo(F)45 b Fp(b)-5 b(e)32 b(a)g(critic)-5 b(al)33 b(emb)-5 b(e)g(dding)33 b(of)f Fo(E)38 b Fp(and)32 b(let)h Fs(\010)e Fp(b)-5 b(e)32 b(a)g(smo)-5 b(oth)34 b(variation)g(of)e Fo(F)13 b Fp(.)0 764 y(Then)33 b(the)g(se)-5 b(c)g(ond)34 b(variation)g(of)f Fo(E)38 b Fp(is)33 b(given)e(by)i(the)g(formula)521 946 y Fo(@)574 913 y Fn(2)614 946 y Fo(E)5 b Fs([\010\()p Fm(\001)p Fo(;)15 b(s;)g(t)p Fs(\)])p 521 987 534 4 v 697 1070 a Fo(@)5 b(s@)g(t)1065 849 y Fi(\014)1065 903 y(\014)1065 958 y(\014)1065 1012 y(\014)1065 1067 y(\014)1095 1126 y Fn(\()p Fk(s;t)p Fn(\)=0)1347 1008 y Fs(=)1443 884 y Fi(Z)1494 1090 y Fk(M)1588 1008 y Fo(P)1646 1022 y Fk(F)1705 1008 y Fm(h)p Fo(V)g(;)15 b(W)e Fm(i)p Fo(dv)s(ol)2137 1023 y Fk(h)2203 1008 y Fs(+)2294 884 y Fi(Z)2345 1090 y Fk(M)2439 1008 y Fo(Q)2511 1022 y Fk(F)2569 1008 y Fs(\()p Fo(V)5 b(;)15 b(W)e Fs(\))p Fo(dv)s(ol)3001 1022 y Fk(g)3043 1008 y Fo(;)-3068 b Fs(\(4.34\))0 1273 y Fp(wher)-5 b(e)43 b Fo(V)61 b Fp(and)43 b Fo(W)53 b Fp(ar)-5 b(e)43 b(the)f(variational)h(ve)-5 b(ctor)42 b(\014elds)g(gener)-5 b(ate)g(d)43 b(by)f Fs(\010)p Fp(.)68 b(Mor)-5 b(e)g(over,)44 b(if)d Fo(P)3309 1287 y Fk(F)3410 1273 y Fo(>)g Fs(0)p Fp(,)0 1386 y(the)36 b(biline)-5 b(ar)36 b(map)h Fo(q)727 1400 y Fk(F)821 1386 y Fp(is)e(p)-5 b(ositive)37 b(de\014nite,)f(and)h Fo(M)j Fs(=)31 b Fo(S)2077 1353 y Fn(1)2116 1386 y Fp(;)37 b(then)f(ther)-5 b(e)36 b(is)g(an)g Fo(\017)31 b(>)f Fs(0)36 b Fp(such)f(that)i(al)5 b(l)20 1476 y Fs(~)0 1499 y Fo(F)38 b Fm(2)25 b(E)230 1513 y Fn(\012)318 1499 y Fp(with)33 b Fm(jj)p Fo(F)h Fm(\000)768 1476 y Fs(~)748 1499 y Fo(F)13 b Fm(jj)869 1518 y Fk(C)924 1500 y Fb(2)989 1499 y Fo(<)25 b(\017)p Fp(,)32 b(satis\014es)h Fo(E)5 b Fs([)p Fo(F)13 b Fs(])27 b Fm(\024)d Fo(E)5 b Fs([)1956 1476 y(~)1935 1499 y Fo(F)14 b Fs(])p Fp(.)0 1679 y Fl(Theorem)34 b(4.3.2)46 b Fp(Consider)29 b(the)f(standar)-5 b(d)31 b(spher)-5 b(e)29 b Fo(S)1934 1646 y Fn(2)1998 1679 y Fm(\032)c Ff(R)2172 1646 y Fn(3)2239 1679 y Fp(with)k(the)f(induc)-5 b(e)g(d)28 b(Euclide)-5 b(an)29 b(metric.)0 1792 y(L)-5 b(et)32 b Fo(F)39 b Fs(:)25 b Fo(S)364 1759 y Fn(2)429 1792 y Fm(!)g Fo(S)606 1759 y Fn(2)671 1792 y Fm(\032)g Ff(R)845 1759 y Fn(3)916 1792 y Fp(b)-5 b(e)32 b(a)g(critic)-5 b(al)33 b(emb)-5 b(e)g(dding)33 b(of)f Fo(E)5 b Fp(.)42 b(If)32 b Fo(q)2230 1806 y Fk(F)2320 1792 y Fp(is)g(p)-5 b(ositive)32 b(semide\014nite,)h(then)f(the)0 1905 y(se)-5 b(c)g(ond)34 b(variation)g(of)f Fo(E)38 b Fp(is)33 b(p)-5 b(ositive)33 b(semide\014nite.)0 2085 y Fs(Before)e(pro)m(ving)f (Theorems)g(4.3.1)i(and)e(4.3.2,)i(w)m(e)f(discuss)e(some)h(of)h(its)f (consequences.)0 2246 y Fl(Example)k(4.3.3)46 b Fp(Curves)33 b(in)f Ff(R)1187 2213 y Fk(n)1234 2246 y Fp(.)42 b Fs(In)29 b(this)h(case)h Fo(q)1828 2260 y Fk(F)1917 2246 y Fs(is)e(p)s(ositiv)m (e)g(de\014nite)h(if)f(and)h(only)f(if)982 2418 y Fo(k)s Fm(j)p Fo(X)7 b Fm(j)1164 2381 y Fn(2)1225 2418 y Fm(\000)20 b Fs(\003)1379 2432 y Fn(2)1418 2418 y Fs(\()p Fo(X)r(;)15 b(X)7 b Fs(\))p Fo(=)p Fm(j)p Fo(F)1828 2381 y Fe(0)1854 2418 y Fm(j)26 b Fo(>)f Fs(0)p Fo(;)198 b(X)32 b Fm(2)25 b Fl(R)2541 2381 y Fk(n)2587 2418 y Fo(:)0 2591 y Fs(T)-8 b(o)31 b(v)m(erify)f(this)f(w)m(e)i(compute)f(form)m(ula)g(\(4)p Fo(:)p Fs(33\))j(using)c(the)h(iden)m(tities)197 2763 y(\()p Fo(I)d Fm(\000)20 b Fo(S)446 2777 y Fk(v)487 2763 y Fs(\))522 2725 y Fe(\000)p Fn(1)632 2763 y Fs(det\()p Fo(I)28 b Fm(\000)19 b Fo(S)1007 2777 y Fk(v)1048 2763 y Fs(\))26 b(=)f(1)p Fo(;)197 b Fm(h)p Fo(\030)1551 2725 y Fn(#)1615 2763 y Fo(;)15 b(\021)1703 2725 y Fn(#)1766 2763 y Fm(i)26 b Fs(=)f Fo(g)1969 2725 y Fe(\003)2009 2763 y Fs(\()p Fo(\030)t(;)15 b(\021)s Fs(\))p Fo(=)p Fm(j)p Fo(F)2352 2725 y Fe(0)2378 2763 y Fm(j)2403 2725 y Fn(2)2442 2763 y Fo(;)107 b Fs(and)2811 2681 y Fm(p)p 2887 2681 53 4 v 82 x Fo(h=)2984 2702 y Fm(p)p 3060 2702 47 4 v 61 x Fo(g)29 b Fs(=)c Fm(j)p Fo(F)3324 2725 y Fe(0)3348 2763 y Fm(j)p Fo(:)0 2935 y Fs(W)-8 b(e)32 b(get)159 3144 y Fo(q)200 3158 y Fk(F)258 3144 y Fs(\()p Fo(X)c Fm(\012)20 b Fo(\030)t(;)15 b(Y)41 b Fm(\012)19 b Fo(\021)s Fs(\))26 b(=)f Fo(k)s Fm(h)p Fo(X)r(;)15 b(Y)22 b Fm(i)p Fo(g)1318 3106 y Fe(\003)1358 3144 y Fs(\()p Fo(\030)t(;)15 b(\021)s Fs(\))22 b Fm(\000)1673 3020 y Fi(Z)1723 3226 y Fn(\012)1774 3237 y Fg(F)1829 3144 y Fm(h)p Fo(X)r(;)15 b(v)s Fm(ih)p Fo(Y)5 b(;)15 b(v)s Fm(i)p Fo(g)2324 3106 y Fe(\003)2367 3144 y Fs(\()p Fo(\030)t(;)g(\021)s Fs(\))p Fm(j)p Fo(F)2665 3106 y Fe(0)2690 3144 y Fm(j)2715 3106 y Fe(\000)p Fn(2)2810 3144 y Fo(p)p Fs(\()p Fo(F)33 b Fs(+)20 b Fo(v)s Fs(\))p Fo(dv)g Fm(j)p Fo(F)3362 3106 y Fe(0)3385 3144 y Fm(j)p Fo(:)0 3368 y Fs(Simplifying)26 b(and)k(writing)e(the)j(result)e(in)g (terms)h(of)h(\003)1906 3382 y Fn(2)1976 3368 y Fs(giv)m(es)701 3540 y Fo(q)742 3554 y Fk(F)801 3540 y Fs(\()p Fo(X)d Fm(\012)20 b Fo(\030)t(;)15 b(Y)40 b Fm(\012)20 b Fo(\021)s Fs(\))26 b(=)f Fo(g)1549 3503 y Fe(\003)1589 3540 y Fs(\()p Fo(\030)t(;)15 b(\021)s Fs(\))1791 3466 y Fi(\000)1834 3540 y Fo(k)s Fm(h)p Fo(X)r(;)g(Y)21 b Fm(i)g(\000)f Fs(\003)2320 3554 y Fn(2)2360 3540 y Fs(\()p Fo(X)r(;)15 b(Y)21 b Fs(\))p Fo(=)p Fm(j)p Fo(F)2762 3503 y Fe(0)2786 3540 y Fm(j)2811 3466 y Fi(\001)2853 3540 y Fo(;)-2878 b Fs(\(4.35\))0 3712 y(from)30 b(whic)m(h)f(the)i(statemen)m(t)h(follo) m(ws.)0 3893 y Fl(Example)i(4.3.4)46 b Fs(Let)37 b Fo(p)f Fs(b)s(e)g(a)h(radial)d(densit)m(y)i(relativ)m(e)h(to)g(a)g(curv)m(e)f (\000.)59 b(F)-8 b(rom)37 b(Example)e(2.4.3,)40 b(it)0 4006 y(follo)m(ws)c(that)i(the)g(co)m(v)-5 b(ariance)39 b(tensor)e Fo(\033)1455 3973 y Fn(2)1452 4033 y Fe(?)1548 4006 y Fs(is)g(a)h(constan)m(t)g(m)m(ultiple)d Fm(j)p Fo(\033)2546 3973 y Fn(2)2543 4033 y Fe(?)2603 4006 y Fm(j)i Fs(of)h(the)f(Euclidean)f(scalar)0 4118 y(pro)s(duct)26 b(restricted)h(to)g(\012)913 4137 y Fk(F)10 b Fn(\()p Fk(u)p Fn(\))1067 4118 y Fs(.)40 b(If)27 b Fo(k)h Fs(=)d(2)p Fo(\031)s Fm(j)p Fo(\033)1571 4085 y Fn(2)1568 4146 y Fe(?)1628 4118 y Fm(j)p Fo(=L)1760 4085 y Fn(2)1760 4145 y Fk(F)1819 4118 y Fs(,)j(then)f(\000)g(is)f(a)h(critical)f(curv)m(e,)j (b)m(y)d(Example)h(4.2.3.)0 4231 y(In)j(this)f(case,)j(w)m(e)e(ha)m(v)m (e)953 4404 y Fo(q)994 4418 y Fk(F)1052 4404 y Fs(\()p Fo(X)e Fm(\012)20 b Fo(\030)t(;)15 b(Y)40 b Fm(\012)20 b Fo(\021)s Fs(\))26 b(=)f Fo(k)s Fm(h)p Fl(T)p Fo(;)15 b(X)7 b Fm(ih)p Fl(T)p Fo(;)15 b(Y)22 b Fm(i)p Fo(g)2373 4366 y Fe(\003)2414 4404 y Fs(\()p Fo(\030)t(;)15 b(\021)s Fs(\))p Fo(:)0 4576 y Fs(T)-8 b(o)36 b(v)m(erify)f(this,)h(use)f(the)g (iden)m(tities)f(\003)1389 4590 y Fn(0)1463 4576 y Fs(=)f(1)p Fo(=L)1719 4590 y Fk(F)1778 4576 y Fo(;)e(\026)1889 4591 y Fe(?)1981 4576 y Fs(=)j(0)p Fo(;)d Fm(j)p Fo(F)2283 4543 y Fe(0)2306 4576 y Fm(j)j Fs(=)g Fo(L)2532 4590 y Fk(F)2590 4576 y Fo(=)p Fs(2)p Fo(\031)40 b Fs(and)34 b Fo(k)j Fs(=)d(2)p Fo(\031)s Fm(j)p Fo(\033)3322 4543 y Fn(2)3319 4603 y Fe(?)3378 4576 y Fm(j)p Fo(=L)3510 4543 y Fn(2)3510 4603 y Fk(F)3570 4576 y Fs(,)0 4689 y(and)c(write)f(\003)472 4703 y Fn(2)542 4689 y Fs(in)g(terms)i(of)f Fo(\033)1060 4656 y Fn(2)1057 4716 y Fe(?)1146 4689 y Fs(in)f(\(4)p Fo(:)p Fs(35\).)43 b(W)-8 b(e)32 b(ha)m(v)m(e)221 4861 y Fo(q)262 4875 y Fk(F)320 4861 y Fs(\()p Fo(X)c Fm(\012)20 b Fo(\030)t(;)15 b(Y)41 b Fm(\012)20 b Fo(\021)s Fs(\))83 b(=)g Fo(g)1184 4823 y Fe(\003)1224 4861 y Fs(\()p Fo(\030)t(;)15 b(\021)s Fs(\))1426 4787 y Fi(\000)1469 4861 y Fo(k)s Fm(h)p Fo(X)r(;)g(Y)22 b Fm(i)e(\000)g Fs(\003)1955 4875 y Fn(0)1995 4861 y Fs(\()p Fo(\033)2085 4823 y Fn(2)2082 4884 y Fe(?)2161 4861 y Fs(+)g Fo(\026)2307 4876 y Fe(?)2386 4861 y Fm(\012)g Fo(\026)2532 4876 y Fe(?)2591 4861 y Fs(\)\()p Fo(X)r(;)15 b(Y)21 b Fs(\))p Fo(=)p Fm(j)p Fo(F)3028 4823 y Fe(0)3053 4861 y Fm(j)3078 4787 y Fi(\001)984 4998 y Fs(=)83 b Fo(g)1184 4961 y Fe(\003)1224 4998 y Fs(\()p Fo(\030)t(;)15 b(\021)s Fs(\))1426 4925 y Fi(\000)1469 4998 y Fs(2)p Fo(\031)s Fm(j)p Fo(\033)1649 4961 y Fn(2)1646 5022 y Fe(?)1706 4998 y Fm(jh)p Fo(X)r(;)g(Y)21 b Fm(i)p Fo(=L)2099 4961 y Fn(2)2099 5021 y Fk(F)2179 4998 y Fm(\000)f Fs(\(1)p Fo(=L)2457 5012 y Fk(F)2517 4998 y Fs(\))p Fo(\033)2607 4961 y Fn(2)2604 5022 y Fe(?)2663 4998 y Fs(\()p Fo(X)r(;)15 b(Y)21 b Fs(\))p Fo(=)p Fs(\()p Fo(L)3066 5012 y Fk(F)3126 4998 y Fo(=)p Fs(2)p Fo(\031)s Fs(\))3306 4925 y Fi(\001)3349 4998 y Fo(:)0 5171 y Fs(Since)41 b(the)i(co)m(v)-5 b(ariance)44 b(tensor)e(is)g(a)g(constan)m(t)i(m)m (ultiple)c Fm(j)p Fo(\033)2173 5138 y Fn(2)2170 5198 y Fe(?)2229 5171 y Fm(j)j Fs(of)g(the)f(Euclidean)f(scalar)h(pro)s (duct)0 5284 y(restricted)f(to)g(\012)603 5302 y Fk(F)10 b Fn(\()p Fk(u)p Fn(\))757 5284 y Fs(,)44 b(w)m(e)d(get)h(that)f(the)g (scalar)g(pro)s(duct)f(of)h(the)g(normal)f(comp)s(onen)m(ts)h Fm(h)p Fo(X)3348 5299 y Fe(?)3407 5284 y Fo(;)15 b(Y)3500 5299 y Fe(?)3559 5284 y Fm(i)0 5396 y Fs(cancels)31 b(\(here)f Fo(X)616 5411 y Fe(?)701 5396 y Fs(=)25 b Fo(X)i Fm(\000)20 b(h)p Fl(T)p Fo(;)15 b(X)7 b Fm(i)p Fl(T)p Fs(\).)42 b(This)29 b(implies)e(the)k(assertion.)p eop %%Page: 37 45 37 44 bop 3803 100 a Fs(37)299 490 y Fl(Example)34 b(4.3.5)46 b Fs(F)-8 b(or)31 b(critical)e(curv)m(es)i(in)e(the)h(plane,)g(w)m(e)h (write)942 695 y Fo(\026)997 710 y Fe(?)1056 695 y Fs(\()p Fl(N)p Fs(\()p Fo(u)p Fs(\)\))26 b(=)f Fo(\026)1507 710 y Fe(?)1566 695 y Fs(\()p Fo(u)p Fs(\))p Fo(;)198 b Fs(and)181 b Fo(\033)2294 658 y Fn(2)2291 718 y Fe(?)2350 695 y Fs(\()p Fl(N)p Fs(\()p Fo(u)p Fs(\))p Fo(;)15 b Fl(N)p Fs(\()p Fo(u)p Fs(\)\))28 b(=)d Fo(\033)3047 658 y Fn(2)3044 718 y Fe(?)3103 695 y Fs(\()p Fo(u)p Fs(\))p Fo(:)299 901 y Fs(Then)k Fo(q)577 915 y Fk(F)666 901 y Fs(is)g(p)s(ositiv)m(e)h (de\014nite)f(if)g(and)h(only)f(if)1610 1106 y Fo(k)g(>)c Fs(2)p Fo(\031)s Fs(\003)1945 1120 y Fn(0)1985 1106 y Fs(\()p Fo(\026)2075 1069 y Fn(2)2075 1129 y Fe(?)2154 1106 y Fs(+)20 b Fo(\033)2300 1069 y Fn(2)2297 1129 y Fe(?)2356 1106 y Fs(\))p Fo(=L)2498 1120 y Fk(F)2557 1106 y Fo(:)299 1312 y Fs(In)27 b(particular,)f(for)i(an)f(ann)m(ular)f (distribution,)f(this)i(condition)f(is)g(satis\014ed)h(if)g(and)g(only) f(if)h(the)h(radius)299 1425 y(of)i(a)h(critical)e(circle)h(is)f(less)h (than)g(1)p Fo(=)p Fs(\(2)p Fo(\031)s(\027)1766 1392 y Fn(0)1807 1425 y Fs(\),)h(where)f Fo(\027)2212 1392 y Fn(0)2281 1425 y Fs(is)g(de\014ned)f(b)m(y)h(\(4)p Fo(:)p Fs(15\).)435 1538 y(T)-8 b(o)34 b(v)m(erify)e(this,)h(w)m(e)h (use)f(the)g(iden)m(tit)m(y)g Fm(j)p Fo(F)1924 1505 y Fe(0)1948 1538 y Fm(j)d Fs(=)f Fo(L)2165 1552 y Fk(F)2224 1538 y Fo(=)p Fs(2)p Fo(\031)s Fs(,)35 b(write)d Fo(X)38 b Fs(=)29 b Fm(h)p Fl(T)p Fo(;)15 b(X)7 b Fm(i)p Fl(T)23 b Fs(+)f Fm(h)p Fl(N)p Fo(;)15 b(X)7 b Fm(i)p Fl(N)p Fs(,)35 b(and)299 1651 y(write)29 b(\003)594 1665 y Fn(2)664 1651 y Fs(in)g(terms)i(of)f Fo(\026)1182 1666 y Fe(?)1271 1651 y Fs(and)g Fo(\033)1503 1618 y Fn(2)1500 1678 y Fe(?)1589 1651 y Fs(in)f(\(4)p Fo(:)p Fs(35\).)43 b(W)-8 b(e)32 b(ha)m(v)m(e)933 1856 y Fo(q)974 1870 y Fk(F)1032 1856 y Fs(\()p Fo(X)c Fm(\012)20 b Fo(\030)t(;)15 b(X)28 b Fm(\012)20 b Fo(\030)t Fs(\))779 1994 y(=)83 b Fo(g)979 1957 y Fe(\003)1019 1994 y Fs(\()p Fo(\030)t(;)15 b(\030)t Fs(\))1217 1920 y Fi(\000)1259 1994 y Fo(k)s Fm(j)p Fo(X)7 b Fm(j)1441 1957 y Fn(2)1502 1994 y Fm(\000)20 b Fs(\003)1656 2008 y Fn(0)1696 1994 y Fs(\()p Fo(\033)1786 1957 y Fn(2)1783 2017 y Fe(?)1862 1994 y Fs(+)g Fo(\026)2008 2009 y Fe(?)2087 1994 y Fm(\012)g Fo(\026)2233 2009 y Fe(?)2292 1994 y Fs(\)\()p Fo(X)r(;)15 b(X)7 b Fs(\))p Fo(=)p Fm(j)p Fo(F)2737 1957 y Fe(0)2763 1994 y Fm(j)2788 1920 y Fi(\001)779 2132 y Fs(=)83 b Fo(g)979 2094 y Fe(\003)1019 2132 y Fs(\()p Fo(\030)t(;)15 b(\030)t Fs(\))1217 2058 y Fi(\000)1259 2132 y Fo(k)s Fs(\()p Fm(h)p Fl(T)p Fo(;)g(X)7 b Fm(i)1609 2094 y Fn(2)1671 2132 y Fs(+)20 b Fm(h)p Fl(N)p Fo(;)15 b(X)7 b Fm(i)2036 2094 y Fn(2)2077 2132 y Fs(\))20 b Fm(\000)g Fs(\003)2286 2146 y Fn(0)2326 2132 y Fs(\()p Fo(\033)2416 2094 y Fn(2)2413 2155 y Fe(?)2492 2132 y Fs(+)g Fo(\026)2638 2094 y Fn(2)2638 2155 y Fe(?)2697 2132 y Fs(\))p Fm(h)p Fl(N)p Fo(;)15 b(X)7 b Fm(i)3006 2094 y Fn(2)3047 2132 y Fo(=)p Fs(\()p Fo(L)3189 2146 y Fk(F)3248 2132 y Fo(=)p Fs(2)p Fo(\031)s Fs(\))3428 2058 y Fi(\001)3471 2132 y Fo(:)299 2337 y Fs(This)31 b(is)h(p)s(ositiv)m(e)g(for)h(all)f Fo(X)37 b Fm(2)29 b Fo(T)1468 2351 y Fk(u)1513 2337 y Fo(S)1574 2304 y Fn(1)1614 2337 y Fo(;)h(X)37 b Fm(6)p Fs(=)30 b(0,)k(if)e(and)g(only)h (if)e Fo(k)i(>)d Fs(2)p Fo(\031)s Fs(\003)2883 2351 y Fn(0)2923 2337 y Fs(\()p Fo(\026)3013 2304 y Fn(2)3013 2365 y Fe(?)3094 2337 y Fs(+)22 b Fo(\033)3242 2304 y Fn(2)3239 2365 y Fe(?)3298 2337 y Fs(\))p Fo(=L)3440 2351 y Fk(F)3499 2337 y Fs(.)48 b(No)m(w)34 b(set)299 2450 y Fo(X)f Fs(=)25 b Fo(d)p Fl(T)p Fo(=ds)30 b Fs(in)f(\(4)p Fo(:)p Fs(6\).)43 b(W)-8 b(e)31 b(get)773 2656 y Fm(\000)p Fs(\003)907 2670 y Fn(1)946 2656 y Fs(\()p Fo(d)p Fl(T)p Fo(=ds)p Fs(\))21 b(+)f(\003)1446 2670 y Fn(2)1486 2656 y Fs(\()p Fo(d)p Fl(T)p Fo(=ds;)15 b(d)p Fl(T)p Fo(=ds)p Fs(\))22 b Fm(\000)e Fs(\()p Fo(k)s(L)2366 2670 y Fk(F)2425 2656 y Fo(=)p Fs(2)p Fo(\031)s Fs(\))p Fm(h)p Fo(d)p Fl(T)p Fo(=ds;)15 b(d)p Fl(T)p Fo(=ds)p Fm(i)28 b Fs(=)d(0)p Fo(:)299 2861 y Fs(By)j(\(4)p Fo(:)p Fs(35\),)i(w)m(e)e(ha)m(v)m(e)h (that)f Fo(q)1297 2875 y Fk(F)1383 2861 y Fs(is)e(p)s(ositiv)m(e)h (de\014nite)f(if)h(and)g(only)f(if)h Fm(\000)p Fs(\003)2791 2875 y Fn(1)2830 2861 y Fs(\()p Fo(d)p Fl(T)p Fo(=ds)p Fs(\))f Fo(>)f Fs(0.)41 b(F)-8 b(or)28 b(a)g(critical)299 2974 y(circle)i(with)f(radius)f Fo(r)33 b Fs(in)c(an)h(ann)m(ular)f (distribution,)e(w)m(e)k(ha)m(v)m(e)h(b)m(y)e(de\014nition)e(of)j(\003) 3216 2988 y Fn(1)3255 2974 y Fs(,)1151 3230 y Fm(\000)p Fs(\003)1285 3244 y Fn(1)1324 3230 y Fs(\()p Fo(d)p Fl(T)p Fo(=ds)p Fs(\))c(=)e Fm(\000)1858 3106 y Fi(Z)1948 3133 y Fk(r)1980 3142 y Fb(2)2015 3133 y Fe(\000)p Fk(r)1908 3313 y(r)1940 3322 y Fb(1)1974 3313 y Fe(\000)p Fk(r)2108 3230 y Fm(h)p Fo(d)p Fl(T)p Fo(=ds;)15 b(v)s Fl(N)p Fm(i)p Fo(p)p 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Fn(0)1352 5191 y Fs(\()p Fo(\026)1442 5206 y Fe(?)1521 5191 y Fm(\012)f Fo(\026)1667 5206 y Fe(?)1746 5191 y Fs(+)g Fo(\033)1892 5153 y Fn(2)1889 5214 y Fe(?)1948 5191 y Fs(\))h(+)e(\003)2157 5205 y Fn(3)2197 5191 y Fs(\()s Fl(^)-55 b(n)p Fs(\))p Fo(=r)2414 5117 y Fi(\001)2456 5191 y Fs(\()p Fo(X)r(;)15 b(Y)22 b Fs(\))p Fo(;)197 b(X)r(;)15 b(Y)46 b Fm(2)25 b Fl(R)3320 5153 y Fk(n)3367 5191 y Fo(;)-3093 b Fs(\(4.36\))299 5396 y(is)29 b(p)s(ositiv)m(e)h(semide\014nite.)39 b(T)-8 b(o)30 b(v)m(erify)g(this)f(claim,)h(w)m(e)h(compute)g Fo(q)2638 5410 y Fk(F)2726 5396 y Fs(and)f(apply)f(Theorem)h(4.3.2.)p eop %%Page: 38 46 38 45 bop 0 100 a Fs(38)136 490 y(W)-8 b(e)32 b(substitute)d(the)i (iden)m(tities)157 685 y Fm(h)p Fo(\030)236 647 y Fn(#)299 685 y Fo(;)15 b(\021)387 647 y Fn(#)451 685 y Fm(i)26 b Fs(=)f Fo(g)654 647 y Fe(\003)694 685 y Fs(\()p Fo(\030)t(;)15 b(\021)s Fs(\))p Fo(=r)985 647 y Fn(2)1026 685 y Fo(;)197 b Fs(\()p Fo(I)27 b Fm(\000)20 b Fo(S)1497 699 y Fk(v)1538 685 y 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y Fo(dv)s(ol)3050 5135 y Fk(h)1371 5360 y Fs(+)p Fo(k)1507 5236 y Fi(Z)1557 5442 y Fk(M)1636 5360 y Fs(\()p Fo(\016)1714 5322 y Fk(ij)1776 5360 y Fm(h)p Fo(dV)21 b Fs(\()p Fo(e)2009 5322 y Fk(M)2009 5382 y(i)2088 5360 y Fs(\))p Fo(;)15 b(dW)e Fs(\()p Fo(e)2386 5322 y Fk(M)2386 5382 y(j)2467 5360 y Fs(\))p Fm(i)21 b Fs(+)f Fm(h)p Fs(\001)2760 5374 y Fk(g)2800 5360 y Fo(F)s(;)15 b(Z)7 b Fm(i)p Fs(\))p Fo(dv)s(ol)3205 5374 y Fk(g)3246 5360 y Fo(:)-3271 b Fs(\(4.37\))p eop %%Page: 39 47 39 46 bop 3803 100 a Fs(39)299 490 y(Since)31 b Fo(F)46 b Fs(is)31 b(a)i(critical)e(em)m(b)s(edding,)h(Theorem)g(4.2.1)i (implies)29 b(that)k(the)g(terms)f(in)m(v)m(olving)f Fo(Z)39 b Fs(cancel.)299 603 y(W)-8 b(riting)29 b(\(4)p Fo(:)p Fs(37\))k(in)c(terms)h(of)h Fo(Q)1430 617 y Fk(F)1519 603 y Fs(establishes)e(\(4)p Fo(:)p Fs(34\).)435 811 y(Next)k(w)m(e)f(pro)m(v)m(e)g(the)f(second)h(statemen)m(t)h(of)f(the)g (theorem;)g(i.e.,)g(the)g(su\016cien)m(t)f(conditions)f(for)h(a)299 924 y(lo)s(cal)f(minim)m(um)d(of)k Fo(E)5 b Fs(.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)h(the)e(follo)m(wing)f(prop)s(osition.)299 1107 y Fl(Prop)s(osition)36 b(4.3.7)46 b Fp(Ther)-5 b(e)46 b(is)f(a)h(neighb)-5 b(orho)g(o)g(d)48 b Fo(U)2214 1121 y Fn(0)2299 1107 y Fp(of)d Fo(F)2476 1121 y Fn(0)2561 1107 y Fp(in)g Fo(C)2757 1074 y Fn(2)2795 1107 y Fs(\()p Fo(S)2891 1074 y Fn(1)2931 1107 y Fo(;)15 b Ff(R)3050 1074 y Fk(n)3097 1107 y Fs(\))45 b Fp(such)g(that)h(for)g(any)299 1220 y(emb)-5 b(e)g(dding)46 b Fo(F)57 b Fp(in)44 b(this)i(neighb)-5 b(orho)g(o)g(d)48 b(satisfying)d Fm(j)p Fo(F)2237 1187 y Fe(0)2260 1220 y Fm(j)2285 1187 y Fe(0)2356 1220 y Fs(=)h(0)p Fp(,)i(ther)-5 b(e)46 b(exists)e Fo(V)67 b Fm(2)47 b Fo(N)3394 1234 y Fk(F)3439 1243 y Fb(0)3522 1220 y Fp(such)d(that)299 1332 y Fo(F)52 b Fs(=)39 b(\()p Fo(F)612 1346 y Fn(0)679 1332 y Fs(+)25 b Fo(V)20 b Fs(\))27 b Fm(\016)f Fo(\036)p Fp(,)42 b(wher)-5 b(e)42 b Fo(\036)d Fs(:)h Fo(S)1589 1299 y Fn(1)1668 1332 y Fm(!)f Fo(S)1859 1299 y Fn(1)1939 1332 y Fp(is)h(a)h(di\013e)-5 b(omorphism)43 b(that)f(viewe)-5 b(d)41 b(as)g(a)g(r)-5 b(e)g(al-value)g(d)299 1445 y(function)31 b(has)i(its)e(inverse)h(de\014ne)-5 b(d)32 b(by)f Fs(\(4)p Fo(:)p Fs(29\))p Fp(.)44 b(Mor)-5 b(e)g(over,)33 b(for)f(al)5 b(l)31 b(such)h(emb)-5 b(e)g(ddings)33 b Fo(F)38 b Fm(2)25 b Fo(U)3601 1459 y Fn(0)3640 1445 y Fp(,)32 b(ther)-5 b(e)299 1558 y(exist)37 b Fo(C)q(;)15 b(\017)34 b(>)f Fs(0)38 b Fp(such)f(that)h(if)f Fm(jj)p Fo(F)g Fm(\000)23 b Fo(F)1670 1572 y Fn(0)1710 1558 y Fm(jj)1760 1578 y Fk(C)1815 1559 y Fb(2)1887 1558 y Fo(<)33 b(\017)p Fp(,)38 b(then)g(the)f(se)-5 b(c)g(ond)39 b(variation)f(of)g Fo(E)k Fp(at)c Fo(F)50 b Fp(satis\014es)299 1671 y Fo(\016)342 1638 y Fn(2)382 1671 y Fo(E)5 b Fs([)p Fo(F)13 b Fs(])26 b Fm(\025)f Fo(C)7 b Fm(jj)p Fo(V)21 b Fm(jj)943 1638 y Fn(2)943 1696 y(1)982 1671 y Fp(,)33 b(wher)-5 b(e)34 b Fm(jj)p Fo(V)20 b Fm(jj)1473 1638 y Fn(2)1473 1696 y(1)1546 1671 y Fp(is)32 b(the)h(squar)-5 b(e)34 b(of)f(the)g(Sob)-5 b(olev)33 b(norm)h(of)f Fo(V)20 b Fp(.)299 1949 y(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)35 b(4.52)p Fs(.)435 2158 y(This)42 b(is)g(a)h(consequence)h(of)f(Lemma)h(4.2.6.)80 b(T)-8 b(o)44 b(pro)m(v)m(e)g(the)f(\\moreo)m(v)m(er")j(part)d(w)m(e)g (let)g Fo(W)59 b Fm(2)299 2271 y Fo(C)371 2238 y Fe(1)445 2271 y Fs(\()p Fo(S)541 2238 y Fn(1)581 2271 y Fo(;)15 b Fl(R)700 2238 y Fk(n)747 2271 y Fs(\))30 b(b)s(e)g(a)h(v)-5 b(ariational)29 b(v)m(ector)j(\014eld.)39 b(W)-8 b(e)32 b(compute)f Fo(\016)2538 2238 y Fn(2)2578 2271 y Fo(E)5 b Fs(\()p Fo(W)m(;)15 b(W)e Fs(\))31 b(as)619 2511 y Fo(\016)662 2473 y Fn(2)703 2511 y Fo(E)5 b Fs(\()p Fo(W)m(;)15 b(W)e Fs(\))84 b(=)1305 2387 y Fi(Z)1355 2593 y Fk(S)1402 2574 y Fb(1)1441 2511 y Fm(f)p Fo(P)1544 2525 y Fk(F)1603 2511 y Fm(h)p Fo(W)1724 2526 y Fe(?)1783 2511 y Fo(;)15 b(W)1909 2526 y Fe(?)1969 2511 y Fm(ij)p Fo(F)2100 2473 y Fe(0)2124 2511 y Fm(j)20 b Fs(+)g Fo(k)s Fs(\()p Fm(h)p Fl(T)p Fo(;)15 b(W)2592 2473 y Fe(0)2579 2537 y(k)2619 2511 y Fo(=)p Fm(j)p Fo(F)2760 2473 y Fe(0)2784 2511 y Fm(ji)21 b(\000)f(h)p Fl(T)3064 2473 y Fe(0)3087 2511 y Fo(=)p Fm(j)p Fo(F)3228 2473 y Fe(0)3253 2511 y Fm(j)p Fo(;)15 b(W)3404 2526 y Fe(?)3463 2511 y Fm(i)p Fs(\))3533 2473 y Fn(2)1487 2710 y Fs(+)p Fo(q)1599 2724 y Fk(F)1656 2710 y Fs(\()p Fm(r)1767 2672 y Fe(?)1827 2710 y Fo(W)1913 2725 y Fe(?)1971 2710 y Fo(;)g Fm(r)2087 2672 y Fe(?)2147 2710 y Fo(W)2233 2725 y Fe(?)2292 2710 y Fs(\))p Fm(g)p Fo(du:)-2197 b Fs(\(4.38\))299 2908 y(W)-8 b(e)32 b(pro)m(v)m(e)f(this) f(form)m(ula)g(later,)h(w)m(e)g(use)g(it)f(no)m(w)h(to)h(\014nd)d(a)i (lo)m(w)m(er)g(b)s(ound)e(of)i Fo(\016)3060 2875 y Fn(2)3100 2908 y Fo(E)5 b Fs(\()p Fo(W)m(;)15 b(W)e Fs(\).)43 b(Since)30 b Fo(q)3811 2922 y Fk(F)3856 2931 y Fb(0)299 3021 y Fs(is)35 b(p)s(ositiv)m(e)g(de\014nite,)h(b)m(y)g(Lemma)g(4.3.10,)k(there)c(are) g Fo(C)2310 3035 y Fn(1)2350 3021 y Fo(;)15 b(\017)2427 3035 y Fn(1)2501 3021 y Fo(>)34 b Fs(0)j(suc)m(h)e(that)i(for)e(all)g Fo(F)48 b Fm(2)34 b(E)3626 3035 y Fn(\012)3717 3021 y Fs(with)299 3134 y Fm(jj)p Fo(F)g Fm(\000)20 b Fo(F)590 3148 y Fn(0)629 3134 y Fm(jj)679 3153 y Fk(C)734 3135 y Fb(2)799 3134 y Fo(<)25 b(\017)932 3148 y Fn(1)971 3134 y Fs(,)31 b(w)m(e)g(ha)m(v)m(e)1461 3332 y Fo(q)1502 3346 y Fk(F)1560 3332 y Fs(\()p Fo(dW)1728 3347 y Fe(?)1788 3332 y Fo(;)15 b(dW)1961 3347 y Fe(?)2020 3332 y Fs(\))26 b Fm(\025)f Fo(C)2242 3346 y Fn(1)2281 3332 y Fm(j)p Fo(W)2405 3294 y Fe(0)2392 3355 y(?)2451 3332 y Fo(=)p Fm(j)p Fo(F)2592 3294 y Fe(0)2617 3332 y Fm(jj)2667 3294 y Fn(2)2706 3332 y Fo(:)-2432 b Fs(\(4.39\))299 3530 y(If)30 b Fo(P)448 3544 y Fk(F)493 3553 y Fb(0)557 3530 y Fo(>)25 b Fs(0,)31 b(b)m(y)f(Lemma)h(4.3.8,)h(there)f(are)f Fo(C)1896 3544 y Fn(2)1936 3530 y Fo(;)15 b(\017)2013 3544 y Fn(2)2078 3530 y Fo(>)25 b Fs(0)30 b(suc)m(h)g(that)h(if)f Fm(jj)p Fo(F)k Fm(\000)19 b Fo(F)3025 3544 y Fn(0)3065 3530 y Fm(jj)3115 3550 y Fk(C)3170 3531 y Fb(2)3235 3530 y Fo(<)25 b(\017)3368 3544 y Fn(2)3407 3530 y Fs(,)31 b(then)1840 3728 y Fo(P)1898 3742 y Fk(F)1957 3728 y Fm(j)p Fo(F)2053 3690 y Fe(0)2077 3728 y Fm(j)25 b(\025)g Fo(C)2288 3742 y Fn(2)2327 3728 y Fo(:)-2053 b Fs(\(4.40\))299 3926 y(Com)m(bining)28 b(the)j(inequalities)c(\(4)p Fo(:)p Fs(39\))33 b(and)d(\(4)p Fo(:)p Fs(40\))i(to)f(estimate)h(\(4)p Fo(:)p Fs(38\),)g(giv)m(es)813 4166 y Fo(\016)856 4129 y Fn(2)896 4166 y Fo(E)5 b Fs(\()p Fo(W)m(;)15 b(W)e Fs(\))27 b Fm(\025)1383 4042 y Fi(Z)1433 4248 y Fk(S)1480 4230 y Fb(1)1518 4166 y Fm(f)p Fo(C)1628 4180 y Fn(2)1668 4166 y Fm(j)p Fo(W)1779 4181 y Fe(?)1838 4166 y Fm(j)1863 4129 y Fn(2)1923 4166 y Fs(+)20 b Fo(C)2079 4180 y Fn(1)2119 4166 y Fm(j)p Fo(W)2243 4129 y Fe(0)2230 4189 y(?)2289 4166 y Fo(=)p Fm(j)p Fo(F)2430 4129 y Fe(0)2454 4166 y Fm(jj)2504 4129 y Fn(2)2544 4166 y Fm(g)p Fo(du)26 b Fm(\025)f Fs(min)n Fm(f)p Fo(C)3071 4180 y Fn(1)3111 4166 y Fo(;)15 b(C)3216 4180 y Fn(2)3256 4166 y Fm(gjj)p Fo(W)3437 4181 y Fe(?)3497 4166 y Fm(jj)3547 4129 y Fn(2)3547 4189 y(1)3587 4166 y Fo(:)-3313 b Fs(\(4.41\))299 4410 y(No)m(w)48 b(w)m(e)h(consider)e(a)h(constan)m(t)h(sp)s(eed)e(v)-5 b(ariation)47 b(\010)2257 4424 y Fk(t)2341 4410 y Fs(=)54 b(\()p Fo(F)46 b Fs(+)31 b Fo(tV)20 b Fs(\))33 b Fm(\016)f Fo(\036)3012 4424 y Fk(t)3090 4410 y Fs(and)47 b(compute)h Fo(W)67 b Fs(=)299 4523 y(\()p Fo(d)p Fs(\010)447 4537 y Fk(t)477 4523 y Fo(=dt)p Fs(\))p Fm(j)662 4537 y Fk(t)p Fn(=0)783 4523 y Fs(.)41 b(By)30 b(the)h(c)m(hain)f(rule,)873 4737 y Fo(d)p Fs(\010)986 4751 y Fk(t)p 873 4777 143 4 v 904 4861 a Fo(dt)1025 4639 y Fi(\014)1025 4694 y(\014)1025 4748 y(\014)1025 4803 y(\014)1025 4857 y(\014)1056 4916 y Fk(t)p Fn(=0)1201 4798 y Fs(=)25 b([\()p Fo(F)1428 4761 y Fe(0)1472 4798 y Fs(+)20 b Fo(tV)1669 4761 y Fe(0)1692 4798 y Fs(\))h Fm(\016)f Fo(\036)1867 4812 y Fk(t)1917 4798 y Fm(\001)h Fo(\036)2017 4761 y Fe(0)2017 4821 y Fk(t)2047 4798 y Fs(])2072 4812 y Fk(t)p Fn(=0)2212 4798 y Fs(+)f Fo(V)40 b Fm(\016)21 b Fo(\036)2516 4812 y Fk(t)2546 4798 y Fm(j)2571 4812 y Fk(t)p Fn(=0)2716 4798 y Fs(=)k Fo(F)2883 4761 y Fe(0)2906 4798 y Fo(\036)2960 4761 y Fe(0)2960 4821 y Fk(t)2990 4798 y Fm(j)3015 4812 y Fk(t)p Fn(=0)3155 4798 y Fs(+)20 b Fo(V)5 b(:)299 5085 y Fs(Since)35 b Fo(F)613 5052 y Fe(0)637 5085 y Fs(\()p Fo(u)p Fs(\))h Fm(2)e Fo(T)943 5099 y Fk(u)989 5085 y Fo(S)1050 5052 y Fn(1)1089 5085 y Fs(,)k(w)m(e)f(get)g Fo(W)1535 5100 y Fe(?)1629 5085 y Fs(=)e Fo(V)21 b Fs(.)58 b(Substituting)34 b(this)h(in)m(to)h(\(4)p Fo(:)p Fs(41\),)41 b(w)m(e)c(get)g(that)g(if)e Fm(jj)p Fo(F)j Fm(\000)299 5198 y Fo(F)357 5212 y Fn(0)397 5198 y Fm(jj)447 5218 y Fk(C)502 5199 y Fb(2)566 5198 y Fo(<)25 b Fs(min)o Fm(f)p Fo(\017)896 5212 y Fn(1)936 5198 y Fo(;)15 b(\017)1013 5212 y Fn(2)1052 5198 y Fm(g)p Fs(,)31 b(then)f(the)h(the)f(second)h(v)-5 b(ariation)29 b Fo(\016)2391 5165 y Fn(2)2432 5198 y Fo(E)5 b Fs(\()p Fo(W)m(;)15 b(W)e Fs(\))31 b(is)f(b)s(ounded)e(b)s(elo)m(w)i(b)m(y)1446 5396 y Fo(\016)1489 5359 y Fn(2)1529 5396 y Fo(E)5 b Fs(\()p Fo(W)m(;)15 b(W)e Fs(\))27 b Fm(\025)e Fs(min)n Fm(f)p Fo(C)2277 5410 y Fn(1)2317 5396 y Fo(;)15 b(C)2422 5410 y Fn(2)2462 5396 y Fm(gjj)p Fo(V)21 b Fm(jj)2681 5359 y Fn(2)2681 5419 y(1)2721 5396 y Fo(:)p eop %%Page: 40 48 40 47 bop 0 100 a Fs(40)0 490 y(This)29 b(pro)m(v)m(es)i(Prop)s (osition)d(4.3.7.)3280 603 y Fl(Q.E.D.)0 763 y Fs(W)-8 b(e)43 b(no)m(w)e(con)m(tin)m(ue)h(with)f(the)h(pro)s(of)e(of)i (Theorem)g(4.3.1.)76 b(W)-8 b(e)43 b(use)e(Prop)s(osition)f(4.3.7)j(to) g(de\014ne)0 876 y(a)c(constan)m(t)h(sp)s(eed)d(v)-5 b(ariation)38 b(of)g Fo(F)1276 890 y Fn(0)1355 876 y Fs(giv)m(en)g(b)m(y)g(\010)1800 890 y Fk(t)1868 876 y Fs(=)h(\()p Fo(F)2071 890 y Fn(0)2137 876 y Fs(+)25 b Fo(tV)20 b Fs(\))26 b Fm(\016)g Fo(\036)2525 890 y Fk(t)2555 876 y Fs(,)40 b(with)d Fo(\036)2889 890 y Fk(t)2958 876 y Fs(giv)m(en)h(b)m(y)g(\(4)p Fo(:)p Fs(29\).)0 989 y(Consider)29 b(the)h(real-v)-5 b(alued)29 b(function)h(de\014ned)f(b)m(y)h Fo(E)5 b Fs(\()p Fo(t)p Fs(\))26 b(=)f Fo(E)5 b Fs([\010)2254 1003 y Fk(t)2284 989 y Fs(].)41 b(By)31 b(T)-8 b(a)m(ylor's)30 b(Theorem,)955 1240 y Fo(E)5 b Fs([)p Fo(F)13 b Fs(])22 b Fm(\000)d Fo(E)5 b Fs([)p Fo(F)1415 1254 y Fn(0)1456 1240 y Fs(])25 b(=)g Fo(E)1674 1202 y Fe(0)1698 1240 y Fs(\(0\))c(+)1925 1116 y Fi(Z)2016 1142 y Fn(1)1976 1322 y(0)2056 1240 y Fs(\(1)g Fm(\000)f Fo(t)p Fs(\))p Fo(E)2388 1202 y Fe(00)2431 1240 y Fs(\()p Fo(t)p Fs(\))p Fo(dt:)0 1485 y Fs(Since)28 b Fo(F)294 1499 y Fn(0)362 1485 y Fs(is)g(a)h(critical)f(em)m(b)s(edding,)g(then)g Fo(E)1589 1452 y Fe(0)1613 1485 y Fs(\(0\))e(=)f(0.)41 b(Since)27 b Fo(E)2268 1452 y Fe(0)q(0)2336 1485 y Fm(\025)e Fo(C)7 b Fm(jj)p Fo(tV)20 b Fm(jj)2710 1452 y Fn(2)2710 1510 y(1)2776 1485 y Fs(=)25 b Fo(C)7 b(t)2977 1452 y Fn(2)3015 1485 y Fm(jj)p Fo(V)21 b Fm(jj)3189 1452 y Fn(2)3189 1510 y(1)3229 1485 y Fs(,)29 b(w)m(e)h(ha)m(v)m(e)1251 1677 y Fo(E)5 b Fs([)p Fo(F)13 b Fs(])21 b Fm(\000)f Fo(E)5 b Fs([)p Fo(F)1711 1691 y Fn(0)1751 1677 y Fs(])26 b Fm(\025)f Fo(C)7 b Fm(jj)p Fo(V)20 b Fm(jj)2143 1640 y Fn(2)2143 1700 y(1)2183 1677 y Fo(=)p Fs(12)p Fo(:)-2343 b Fs(\(4.42\))0 1882 y(W)-8 b(e)31 b(use)f(this)g(inequalit)m(y)e(to)j (sho)m(w)f(that)h Fo(F)1503 1896 y Fn(0)1573 1882 y Fs(is)f(a)g(lo)s (cal)g(minim)m(um.)38 b(An)m(y)2611 1859 y(~)2591 1882 y Fo(F)g Fm(2)25 b Fo(U)2835 1896 y Fn(0)2905 1882 y Fs(can)31 b(b)s(e)e(written)h(as)20 1972 y(~)0 1995 y Fo(F)38 b Fs(=)25 b(\()p Fo(F)285 2009 y Fn(0)345 1995 y Fs(+)19 b Fo(V)h Fs(\))f Fm(\016)639 1971 y Fs(~)627 1995 y Fo(\036)30 b Fs(with)e Fo(V)46 b Fm(2)25 b Fo(N)1175 2009 y Fk(F)1220 2018 y Fb(0)1288 1995 y Fs(and)1476 1971 y(~)1464 1995 y Fo(\036)g Fs(:)h Fo(S)1655 1962 y Fn(1)1720 1995 y Fm(!)f Fo(S)1897 1962 y Fn(1)1966 1995 y Fs(a)30 b(di\013eomorphism.)38 b(Let)30 b Fo(F)43 b Fs(b)s(e)29 b(the)h(constan)m(t)0 2108 y(sp)s(eed)h(em)m(b)s(edding)g (de\014ned)g(b)m(y)h Fo(F)42 b Fs(=)28 b(\()p Fo(F)1449 2122 y Fn(0)1511 2108 y Fs(+)21 b Fo(V)f Fs(\))i Fm(\016)g Fo(\036)32 b Fs(with)f Fo(\036)i Fs(giv)m(en)f(b)m(y)g(\(4)p Fo(:)p Fs(29\).)49 b(W)-8 b(e)33 b(sho)m(w)g(\014rst)e(that)0 2221 y Fo(E)5 b Fs([)p Fo(F)13 b Fs(])26 b Fm(\024)f Fo(E)5 b Fs([)433 2198 y(~)412 2221 y Fo(F)14 b Fs(],)31 b(and)e(then)h(use)h(\(4)p Fo(:)p Fs(42\))h(to)f(conclude)f(that)h Fo(E)5 b Fs([)p Fo(F)2205 2235 y Fn(0)2245 2221 y Fs(])26 b Fm(\024)f Fo(E)5 b Fs([)2510 2198 y(~)2489 2221 y Fo(F)14 b Fs(])30 b(for)g(all)2902 2198 y(~)2882 2221 y Fo(F)38 b Fm(2)25 b Fo(U)3126 2235 y Fn(0)3165 2221 y Fs(.)136 2334 y(Let)36 b Fo(\015)k Fs(=)33 b(\()p Fo(F)588 2348 y Fn(0)652 2334 y Fs(+)24 b Fo(V)c Fs(\)\()p Fo(S)951 2301 y Fn(1)991 2334 y Fs(\))36 b(b)s(e)f(the)h(image)f(curv)m(e)h(of)g Fo(S)2038 2301 y Fn(1)2113 2334 y Fs(under)e Fo(F)2434 2348 y Fn(0)2498 2334 y Fs(+)23 b Fo(V)d Fs(.)57 b(Since)34 b(the)i(length)f(of)h Fo(\015)0 2447 y Fs(do)s(es)30 b(not)h(dep)s(end)d(on)j(the)f(parameterization)h(of)f Fo(S)1832 2414 y Fn(1)1872 2447 y Fs(,)g(w)m(e)h(ha)m(v)m(e)1178 2568 y Fi(Z)1269 2594 y 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5353 y Fs(\)\))26 b(=)f Fm(h)p Fl(T)p Fo(;)15 b(W)3231 5315 y Fe(0)3218 5380 y(k)3258 5353 y Fo(=)p Fm(j)p Fo(F)3399 5315 y Fe(0)3423 5353 y Fm(ji)3483 5315 y Fn(2)3523 5353 y Fo(:)-3548 b Fs(\(4.44\))p eop %%Page: 41 49 41 48 bop 3803 100 a Fs(41)299 490 y(Since)31 b Fo(W)624 508 y Fe(k)693 490 y Fs(=)d Fm(h)p Fl(T)p Fo(;)15 b(W)1026 508 y Fe(k)1066 490 y Fm(i)p Fl(T)p Fs(,)33 b(then)g Fo(W)1541 457 y Fe(0)1528 521 y(k)1596 490 y Fs(=)28 b Fm(h)p Fl(T)p Fo(;)15 b(W)1929 508 y Fe(k)1969 490 y Fm(i)2004 457 y Fe(0)2028 490 y Fl(T)21 b Fs(+)h Fm(h)p Fl(T)p Fo(;)15 b(W)2449 508 y Fe(k)2488 490 y Fm(i)p Fl(T)2596 457 y Fe(0)2620 490 y Fs(.)47 b(Hence,)34 b(the)f(normal)e (comp)s(onen)m(t)299 616 y(of)f Fo(W)501 583 y Fe(0)488 647 y(k)558 616 y Fs(is)f(giv)m(en)i(b)m(y)1510 856 y Fo(B)5 b Fs(\()p Fo(e)1661 818 y Fk(M)1661 878 y(i)1740 856 y Fo(;)15 b(W)1866 874 y Fe(k)1906 856 y Fs(\))25 b(=)g Fm(h)p Fl(T)p Fo(;)15 b(W)2296 874 y Fe(k)2336 856 y Fm(i)p Fl(T)2444 818 y Fe(0)2467 856 y Fo(=)p 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Fm(j)3093 3284 y Fi(\013)3136 3357 y Fo(:)-2862 b Fs(\(4.48\))299 3571 y(By)31 b(\(4)p Fo(:)p Fs(46\),)961 3784 y Fm(\000h)p Fo(dW)1200 3799 y Fe(?)1259 3784 y Fs(\()p Fo(W)1380 3803 y Fe(k)1419 3784 y Fs(\))p Fo(;)15 b(B)5 b Fs(\()p Fo(e)1645 3747 y Fk(M)1645 3807 y(i)1725 3784 y Fo(;)15 b(e)1807 3747 y Fk(M)1807 3807 y(j)1887 3784 y Fs(\))p Fm(i)26 b Fs(=)f Fm(\000)2150 3711 y Fi(\012)2193 3784 y Fm(h)p Fl(T)p Fo(;)15 b(W)2427 3803 y Fe(k)2467 3784 y Fm(i)p Fo(W)2601 3747 y Fe(0)2588 3807 y(?)2647 3784 y Fo(=)p Fm(j)p Fo(F)2788 3747 y Fe(0)2812 3784 y Fm(j)p Fo(;)g Fl(T)2950 3747 y Fe(0)2974 3784 y Fo(=)p Fm(j)p Fo(F)3115 3747 y Fe(0)3139 3784 y Fm(j)3164 3711 y Fi(\013)3207 3784 y Fo(:)-2933 b Fs(\(4.49\))299 3998 y(By)38 b(Example)f(2.1.2,)42 b(the)c(shap)s(e)f(op)s(erator)h Fo(S)1943 4012 y Fk(W)2010 4024 y Fa(?)2103 3998 y Fs(is)f(an)h(scalar)g(m)m(ultiplication)c(b)m (y)k Fm(h)p Fl(T)3434 3965 y Fe(0)3457 3998 y Fo(=)p Fm(j)p Fo(F)3598 3965 y Fe(0)3623 3998 y Fm(j)p Fo(;)15 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5422 y Fa(?)1192 5396 y Fo(e)1234 5359 y Fk(M)1234 5419 y(j)1313 5396 y Fm(i)26 b Fs(=)1470 5323 y Fi(\012)1513 5396 y Fm(h)p Fl(T)1621 5359 y Fe(0)1644 5396 y Fo(=)p Fm(j)p Fo(F)1785 5359 y Fe(0)1809 5396 y Fm(j)p Fo(;)15 b(W)1960 5411 y Fe(?)2020 5396 y Fm(i)p Fl(T)p Fo(;)g Fm(h)p Fl(T)2276 5359 y Fe(0)2300 5396 y Fo(=)p Fm(j)p Fo(F)2441 5359 y Fe(0)2465 5396 y Fm(j)p Fo(;)g(W)2616 5411 y Fe(?)2675 5396 y Fm(i)p Fl(T)2783 5323 y Fi(\013)2852 5396 y Fs(=)25 b Fm(h)p Fl(T)3056 5359 y Fe(0)3079 5396 y Fo(=)p Fm(j)p Fo(F)3220 5359 y Fe(0)3244 5396 y Fm(j)p Fo(;)15 b(W)3395 5411 y Fe(?)3455 5396 y Fm(i)3490 5359 y Fn(2)3529 5396 y Fo(:)p eop %%Page: 42 50 42 49 bop 0 100 a Fs(42)0 490 y(Substituting)28 b(this)h(in)m(to)h(\(4) p Fo(:)p Fs(63\))j(giv)m(es)426 732 y Fo(\016)469 694 y Fn(2)509 732 y Fo(E)5 b Fs(\()p Fo(W)702 747 y Fe(?)762 732 y Fo(;)15 b(W)888 747 y Fe(?)947 732 y Fs(\))83 b(=)1219 608 y Fi(Z)1270 814 y Fk(S)1317 796 y Fb(1)1370 732 y Fo(P)1428 746 y Fk(F)1487 732 y Fm(h)p Fo(W)1608 747 y Fe(?)1667 732 y Fo(;)15 b(W)1793 747 y Fe(?)1853 732 y Fm(ij)p Fo(F)1984 694 y Fe(0)2008 732 y Fm(j)p Fo(du)1219 971 y Fs(+)1305 848 y Fi(Z)1355 1054 y Fk(S)1402 1035 y Fb(1)1441 971 y Fm(f)p Fo(k)s Fm(h)p Fl(T)1644 934 y Fe(0)1668 971 y Fo(=)p Fm(j)p Fo(F)1809 934 y Fe(0)1833 971 y Fm(j)p Fo(;)g(W)1984 986 y Fe(?)2043 971 y Fm(i)2078 934 y Fn(2)2138 971 y Fs(+)20 b Fo(q)2270 985 y Fk(F)2329 971 y Fs(\()p Fm(r)2440 934 y Fe(?)2499 971 y Fo(W)2585 986 y Fe(?)2644 971 y Fo(;)15 b Fm(r)2760 934 y Fe(?)2819 971 y Fo(W)2905 986 y Fe(?)2964 971 y Fs(\))p Fm(g)p Fo(du:)-3168 b Fs(\(4.52\))0 1225 y(No)m(w)28 b(w)m(e)g(add)f(\(4)p Fo(:)p Fs(47\),)j(t)m(wice)e(\(4)p Fo(:)p Fs(51\))i(and)c(\(4)p Fo(:)p Fs(52\))k(to)e(obtain)f Fo(\016)2150 1192 y Fn(2)2190 1225 y Fo(E)5 b Fs(\()p Fo(W)m(;)15 b(W)e Fs(\).)41 b(After)28 b(simplifying,)c(w)m(e)k(get)0 1338 y(\(4)p Fo(:)p Fs(38\).)0 1548 y Fp(Pr)-5 b(o)g(of)34 b(of)f(The)-5 b(or)g(em)34 b(4.3.2)136 1758 y Fs(W)-8 b(e)44 b(compute)f 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Fe(?)706 2248 y Fo(V)759 2263 y Fe(?)847 2248 y Fs(=)e Fl(^)-55 b(n)21 b Fm(\012)h Fo(d)-15 b(f)10 b Fs(.)46 b(Since)31 b Fo(q)1556 2262 y Fk(F)1647 2248 y Fs(is)g(p)s(ositiv)m(e)g(de\014nite,)h(w)m(e)h(ha)m(v)m(e)h Fo(q)2818 2262 y Fk(F)2876 2248 y Fs(\()p Fm(r)2987 2215 y Fe(?)3046 2248 y Fo(V)3099 2263 y Fe(?)3158 2248 y Fo(;)15 b Fm(r)3274 2215 y Fe(?)3334 2248 y Fo(V)3387 2263 y Fe(?)3446 2248 y Fs(\))29 b(=)f Fo(q)3650 2262 y Fk(F)3708 2248 y Fs(\()s Fl(^)-55 b(n)22 b Fm(\012)299 2361 y Fo(d)-15 b(f)5 b(;)18 b Fl(^)-55 b(n)15 b Fm(\012)g Fo(d)-15 b(f)10 b Fs(\))24 b Fm(\025)h Fs(0.)40 b(Therefore,)28 b(b)m(y)g(\(4)p Fo(:)p Fs(63\),)j(w)m(e)d(ha)m(v)m(e)h Fo(\016)2158 2328 y Fn(2)2198 2361 y Fo(E)5 b Fs(\()p Fo(V)2358 2376 y Fe(?)2418 2361 y Fo(;)15 b(V)2511 2376 y Fe(?)2570 2361 y Fs(\))26 b Fm(\025)f Fs(0.)40 b(This)26 b(sho)m(ws)h(Theorem)h(4.3.2.)435 2568 y(W)-8 b(e)37 b(conclude)e(this)f(section)i(with)e(some)h(tec)m(hnical)h(lemmas)f (used)f(in)g(the)i(pro)s(ofs)f(of)g(Theorems)299 2681 y(4.3.1)d(and)e(Theorem)g(4.43.)299 2978 y Fl(Lemma)j(4.3.8)46 b Fp(The)33 b(map)h Fo(P)k Fs(:)25 b Fm(E)1510 2992 y Fn(\012)1586 2978 y Fm(\002)20 b Fo(C)7 b Fs(\(\012\))25 b Fm(!)g Fo(C)7 b Fs(\()p Fo(M)j Fs(\))33 b Fp(given)f(by)1191 3215 y Fo(P)13 b Fs(\()p Fo(F)s(;)i(p)p Fs(\))26 b(=)1601 3092 y Fi(Z)1652 3298 y Fn(\012)1703 3309 y Fg(F)1773 3215 y Fo(p)p Fs(\()p Fo(F)33 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2510 3229 y Fk(v)2550 3215 y Fs(\))p Fo(dv)2679 3134 y Fm(p)p 2756 3134 53 4 v 2756 3215 a Fo(h=)2853 3154 y Fm(p)p 2930 3154 47 4 v 2930 3215 a Fo(g)s(;)299 3468 y Fp(is)36 b(c)-5 b(ontinuous.)55 b(Her)-5 b(e)36 b(we)h(c)-5 b(onsider)38 b Fo(C)7 b Fs(\()p Fo(M)j Fs(\))36 b Fp(and)i Fo(C)7 b Fs(\(\012\))36 b Fp(as)h(Banach)h(sp)-5 b(ac)g(es)38 b(with)f(the)g(norm)h(of)e(the)299 3581 y(supr)-5 b(eme,)33 b(and)h(we)f(also)h(c)-5 b(onsider)33 b Fm(E)1587 3595 y Fn(\012)1675 3581 y Fp(as)g(a)g(subset)g(of)g Fo(C)2319 3548 y Fn(2)2358 3581 y Fs(\()p Fo(M)5 b(;)15 b Ff(R)2604 3548 y Fk(n)2651 3581 y Fs(\))p Fp(.)299 3785 y(Pr)-5 b(o)g(of)p Fs(.)52 b(Let)35 b Fo(\017)c(>)f Fs(0.)52 b(W)-8 b(e)35 b(write)e Fo(P)13 b Fs(\()p Fo(F)s(;)i(p)p Fs(\))35 b(in)d(terms)i(of)g(a)g(con)m(tin)m(uous)g(function)e (de\014ned)h(on)h Fo(J)3639 3752 y Fn(2)3630 3811 y(\012)3685 3785 y Fs(.)51 b(Let)299 3897 y Fo(s)342 3911 y Fk(p)406 3897 y Fs(:)26 b Fo(J)516 3864 y Fn(2)507 3924 y(\012)588 3897 y Fm(!)f Fl(R)31 b Fs(b)s(e)e(giv)m(en)i(b)m(y)779 4139 y Fo(s)822 4153 y Fk(p)862 4139 y Fs(\()p Fo(j)939 4101 y Fn(2)934 4161 y Fk(x)979 4139 y Fo(F)13 b Fs(\))26 b(=)1207 4015 y Fi(Z)1257 4221 y Fn(\012)1308 4235 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))1461 4139 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2321 4153 y Fk(v)2361 4139 y Fs(\()p Fo(j)2438 4101 y Fn(2)2433 4161 y Fk(x)2479 4139 y Fo(F)13 b Fs(\)\))p Fo(dv)2714 4056 y Fi(p)p 2806 4056 277 4 v 2806 4139 a Fo(h)p Fs(\()p Fo(j)2935 4113 y Fn(2)2930 4161 y Fk(x)2976 4139 y Fo(F)g Fs(\))p Fo(=)3127 4057 y Fi(p)p 3219 4057 170 4 v 3219 4139 a Fo(g)s Fs(\()p Fo(x)p Fs(\))q Fo(:)299 4416 y Fs(Then)28 b(w)m(e)h(write)899 4343 y Fi(\000)940 4416 y Fo(P)13 b Fs(\()p Fo(F)s(;)i(p)p Fs(\))1228 4343 y Fi(\001)1271 4416 y Fs(\()p Fo(x)p Fs(\))26 b(=)f Fo(s)1558 4430 y Fk(p)1597 4416 y Fs(\()p Fo(j)1674 4383 y Fn(2)1669 4439 y Fk(x)1714 4416 y Fo(F)13 b Fs(\).)41 b(Let)29 b(a)h(neigh)m(b)s(orho)s(o)s(d)c Fo(N)36 b Fm(\032)25 b Fo(J)2955 4383 y Fn(2)2946 4443 y(\012)3030 4416 y Fs(of)k(the)g(set)h Fm(f)p Fo(j)3515 4383 y Fn(2)3510 4439 y Fk(x)3556 4416 y Fo(F)38 b Fm(2)25 b Fo(J)3797 4383 y Fn(2)3788 4443 y(\012)3868 4416 y Fs(:)299 4529 y Fo(x)k Fm(2)f Fo(M)10 b Fm(g)33 b Fs(suc)m(h)f(that)h(its)f(closure)p 1486 4456 83 4 v 32 w Fo(N)43 b Fs(is)31 b(compact.)48 b(Since)32 b Fo(s)2383 4543 y Fk(p)2455 4529 y Fs(is)f(uniformly)f(con) m(tin)m(uous)i(on)p 3550 4456 V 32 w Fo(N)10 b Fs(,)33 b(there)299 4642 y(exists)d Fo(\016)f(>)c Fs(0)31 b(suc)m(h)f(that)h (for)f(all)1479 4619 y(~)1458 4642 y Fo(F)38 b Fm(2)25 b(E)1688 4656 y Fn(\012)1774 4642 y Fs(with)k Fm(jj)p Fo(F)34 b Fm(\000)2234 4619 y Fs(~)2213 4642 y Fo(F)14 b Fm(jj)2335 4662 y Fk(C)2390 4643 y Fb(2)2454 4642 y Fo(<)25 b(\016)34 b Fs(w)m(e)d(ha)m(v)m(e)1496 4837 y(sup)1527 4916 y Fk(M)1648 4837 y Fm(j)p Fo(s)1716 4851 y Fk(p)1756 4837 y Fs(\()p Fo(j)1833 4800 y Fn(2)1828 4860 y Fk(x)1873 4837 y Fo(F)13 b Fs(\))21 b Fm(\000)f Fo(s)2134 4851 y Fk(p)2173 4837 y Fs(\()p Fo(j)2250 4800 y Fn(2)2245 4860 y Fk(x)2311 4814 y Fs(~)2290 4837 y Fo(F)14 b Fs(\))p Fm(j)25 b Fo(<)g(\017=)p Fs(2)p Fo(:)-2396 b Fs(\(4.56\))299 5091 y(Consider)28 b(no)m(w)j(the)f(con)m(tin)m(uous)g(function)f Fo(s)c Fs(:)h Fo(J)2013 5058 y Fn(2)2004 5118 y(\012)2085 5091 y Fm(!)f Fl(R)31 b Fs(giv)m(en)f(b)m(y)1041 5333 y Fo(s)p Fs(\()p Fo(j)1161 5295 y Fn(2)1156 5355 y Fk(x)1201 5333 y Fo(F)13 b Fs(\))26 b(=)1429 5209 y Fi(Z)1479 5415 y Fn(\012)1530 5429 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))1683 5333 y Fs(det\()p Fo(I)28 b Fm(\000)20 b Fo(S)2059 5347 y Fk(v)2100 5333 y Fs(\()p Fo(j)2177 5295 y Fn(2)2172 5355 y Fk(x)2217 5333 y Fo(F)13 b Fs(\)\))p Fo(dv)2452 5250 y Fi(p)p 2544 5250 277 4 v 2544 5333 a Fo(h)p Fs(\()p Fo(j)2673 5306 y Fn(2)2668 5355 y Fk(x)2714 5333 y Fo(F)g Fs(\))q Fo(=)2866 5251 y Fi(p)p 2957 5251 170 4 v 82 x Fo(g)s Fs(\()p Fo(x)p Fs(\))q Fo(:)p eop %%Page: 44 52 44 51 bop 0 100 a Fs(44)0 490 y(W)-8 b(e)32 b(see)f(that)g Fo(s)544 504 y Fk(p)608 490 y Fm(\024)25 b Fo(s)p Fm(jj)p Fo(p)p Fm(jj)893 508 y Fk(C)5 b Fn(\(\012\))1058 490 y Fs(.)41 b(Since)29 b Fo(P)44 b Fs(is)29 b(linear)g(in)g Fo(p)p Fs(,)h(w)m(e)h(estimate)g Fm(j)p Fo(P)13 b Fs(\()p Fo(F)s(;)i(p)p Fs(\))21 b Fm(\000)f Fo(P)13 b Fs(\()3064 467 y(~)3043 490 y Fo(F)h(;)23 b Fs(~)-53 b Fo(p)p Fs(\))p Fm(j)31 b Fs(as)297 701 y Fm(j)p Fo(P)13 b Fs(\()p Fo(F)s(;)i(p)p Fs(\))21 b Fm(\000)f Fo(P)13 b Fs(\()849 678 y(~)828 701 y Fo(F)g(;)24 b Fs(~)-54 b Fo(p)q Fs(\))p Fm(j)83 b(\024)g(j)p Fo(P)13 b Fs(\()p Fo(F)s(;)i(p)p Fs(\))21 b Fm(\000)f Fo(P)13 b Fs(\()1835 678 y(~)1814 701 y Fo(F)h(;)h(p)p Fs(\))p Fm(j)21 b Fs(+)e Fo(P)13 b Fs(\()2270 678 y(~)2249 701 y Fo(F)h(;)h(p)20 b Fm(\000)28 b Fs(~)-53 b Fo(p)p Fs(\))1129 839 y Fm(\024)83 b Fs(sup)1314 917 y Fk(M)1435 839 y Fm(j)p Fo(s)1503 853 y Fk(p)1543 839 y Fs(\()p Fo(j)1620 801 y Fn(2)1615 861 y Fk(x)1660 839 y Fo(F)13 b Fs(\))21 b Fm(\000)e Fo(s)1920 853 y Fk(p)1960 839 y Fs(\()p Fo(j)2037 801 y Fn(2)2032 861 y Fk(x)2098 816 y Fs(~)2077 839 y Fo(F)13 b Fs(\))p Fm(j)21 b Fs(+)f(sup)2351 917 y Fk(M)2472 839 y Fo(s)p Fs(\()p Fo(j)2592 801 y Fn(2)2587 861 y Fk(x)2632 839 y Fo(F)13 b Fs(\))21 b Fm(\001)f(jj)p Fo(p)h Fm(\000)27 b Fs(~)-52 b Fo(p)o Fm(jj)3107 857 y Fk(C)5 b Fn(\(\012\))3273 839 y Fo(:)0 1102 y Fs(If)25 b Fm(jj)p Fo(F)e Fm(\000)318 1079 y Fs(~)298 1102 y Fo(F)13 b Fm(jj)419 1121 y Fk(C)474 1103 y Fb(2)538 1102 y Fo(<)25 b(\016)k Fs(and)c Fm(jj)p Fo(p)10 b Fm(\000)17 b Fs(~)-52 b Fo(p)o Fm(jj)1157 1120 y Fk(C)5 b Fn(\(\012\))1348 1102 y Fo(<)25 b(\017=)p Fs(\(2)15 b(sup)p 1759 1076 68 3 v 29 x Fk(N)1842 1102 y Fo(s)p Fs(\),)26 b(b)m(y)f(\(4)p Fo(:)p Fs(56\))j(w)m(e)d(ha)m(v)m(e)i Fm(j)p Fo(P)13 b Fs(\()p Fo(F)s(;)i(p)p Fs(\))10 b Fm(\000)g Fo(P)j Fs(\()3214 1079 y(~)3193 1102 y Fo(F)h(;)23 b Fs(~)-53 b Fo(p)p Fs(\))p Fm(j)26 b Fo(<)f(\017)p Fs(.)0 1215 y(This)k(pro)m(v)m(es)i(the)f(lemma.)0 1396 y Fl(Lemma)j(4.3.9)46 b Fp(L)-5 b(et)26 b Fo(F)38 b Fm(2)25 b(E)1011 1410 y Fn(\012)1091 1396 y Fp(such)g(that)i Fo(q)1510 1410 y Fk(F)1593 1396 y Fp(is)e(p)-5 b(ositive)26 b(de\014nite.)40 b(Ther)-5 b(e)26 b(exists)f Fo(C)32 b(>)25 b Fs(0)h Fp(such)f(that)h (al)5 b(l)0 1509 y Fo(x)25 b Fm(2)g Fo(M)38 b Fp(and)28 b(al)5 b(l)28 b Fs(\()p Fo(X)r(;)15 b(\030)t Fs(\))27 b Fm(2)e Ff(R)1007 1476 y Fk(n)1063 1509 y Fm(\002)9 b Fo(T)1209 1476 y Fe(\003)1196 1531 y Fk(x)1249 1509 y Fo(M)c(;)15 b(X)33 b Fm(6)p Fs(=)25 b(0)p Fo(;)15 b(\030)30 b Fm(6)p Fs(=)25 b(0)p Fp(,)k(satis\014es)f Fo(q)2316 1523 y Fk(F)2374 1509 y Fs(\()p Fo(X)17 b Fm(\012)9 b Fo(\030)t(;)15 b(X)i Fm(\012)9 b Fo(\030)t Fs(\))26 b Fo(>)f(C)7 b(g)3156 1476 y Fe(\003)3196 1509 y Fs(\()p Fo(\030)t(;)15 b(\030)t Fs(\))p Fm(j)p Fo(X)7 b Fm(j)3526 1476 y Fn(2)3567 1509 y Fp(.)0 1690 y(Pr)-5 b(o)g(of)p Fs(.)67 b(Supp)s(ose)37 b(that)j(the)e(lemma)h(is)f(false.)65 b(Then,)40 b(for)f(all)e Fo(n)i Fm(2)g Fl(N)f Fs(there)h(exists)g Fo(x)3126 1704 y Fk(n)3212 1690 y Fm(2)g Fo(M)48 b Fs(and)0 1803 y(\()p Fo(X)110 1817 y Fk(n)158 1803 y Fo(;)15 b(\030)238 1817 y Fk(n)285 1803 y Fs(\))25 b Fm(2)g Fl(R)510 1770 y Fk(n)573 1803 y Fm(\002)16 b Fo(T)726 1770 y Fe(\003)713 1826 y Fk(x)753 1834 y Fg(n)799 1803 y Fo(M)38 b Fs(with)28 b Fm(j)p Fo(X)1231 1817 y Fk(n)1278 1803 y Fm(j)1303 1770 y Fn(2)1368 1803 y Fs(=)d(1)p Fo(;)31 b(g)1611 1770 y Fe(\003)1651 1803 y Fs(\()p Fo(\030)t(;)15 b(\030)t Fs(\))26 b(=)f(1,)30 b(suc)m(h)e(that)h Fo(q)2510 1817 y Fk(F)2568 1803 y Fs(\()p Fo(X)2678 1817 y Fk(n)2742 1803 y Fm(\012)15 b Fo(\030)2868 1817 y Fk(n)2915 1803 y Fo(;)g(X)3030 1817 y Fk(n)3094 1803 y Fm(\012)h Fo(\030)3221 1817 y Fk(n)3268 1803 y Fs(\))25 b Fm(\024)g Fs(1)p Fo(=n)p Fs(.)0 1916 y(Since)i Fo(M)38 b Fs(is)27 b(compact,)i(there)g(exist)e (sequences)h Fm(f)p Fo(x)1786 1930 y Fk(n)1834 1916 y Fm(g)d(\032)g Fo(M)5 b(;)31 b Fm(f)p Fs(\()p Fo(X)2304 1930 y Fk(n)2352 1916 y Fo(;)15 b(\030)2432 1930 y Fk(n)2479 1916 y Fs(\))p Fm(g)26 b(\032)f Fl(R)2760 1883 y Fk(n)2822 1916 y Fm(\002)15 b Fo(T)2974 1883 y Fe(\003)2961 1939 y Fk(x)3001 1947 y Fg(n)3047 1916 y Fo(M)38 b Fs(con)m(v)m(erging)0 2029 y(resp)s(ectiv)m(ely)30 b(to)h Fo(x)25 b Fm(2)g Fo(M)5 b(;)30 b Fs(\()p Fo(X)r(;)15 b(\030)t Fs(\))27 b Fm(2)e Fl(R)1340 1996 y Fk(n)1408 2029 y Fm(\002)20 b Fo(T)1565 1996 y Fe(\003)1552 2052 y Fk(x)1604 2029 y Fo(M)40 b Fs(with)29 b Fm(j)p Fo(X)7 b Fm(j)2071 1996 y Fn(2)2137 2029 y Fs(=)25 b(1)p Fo(;)31 b(g)2380 1996 y Fe(\003)2420 2029 y Fs(\()p Fo(\030)t(;)15 b(\030)t Fs(\))26 b(=)f(1.)41 b(Hence,)540 2226 y Fo(q)581 2240 y Fk(F)640 2226 y Fs(\()p Fo(X)28 b Fm(\012)20 b Fo(\030)t(;)15 b(X)28 b Fm(\012)19 b Fo(\030)t Fs(\))26 b(=)54 b(lim)1347 2280 y Fk(n)p Fe(!1)1546 2226 y Fo(q)1587 2240 y Fk(F)1645 2226 y Fs(\()p Fo(X)1755 2240 y Fk(n)1823 2226 y Fm(\012)20 b Fo(\030)1954 2240 y Fk(n)2001 2226 y Fo(;)15 b(X)2116 2240 y Fk(n)2184 2226 y Fm(\012)20 b Fo(\030)2315 2240 y Fk(n)2361 2226 y Fs(\))26 b Fm(\024)54 b Fs(lim)2518 2280 y Fk(n)p Fe(!1)2717 2226 y Fs(1)p Fo(=n)26 b Fs(=)f(0)p Fo(:)0 2445 y Fs(This)37 b(is)h(a)h(con)m(tradiction)g(since)f(w)m(e)h (assumed)f(that)h Fo(q)1954 2459 y Fk(F)2051 2445 y Fs(w)m(as)h(p)s (ositiv)m(e)d(de\014nite.)65 b(This)37 b(pro)m(v)m(es)j(the)0 2558 y(lemma.)0 2739 y Fl(Lemma)33 b(4.3.10)46 b Fp(L)-5 b(et)30 b Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))27 b Fm(2)e(E)1305 2753 y Fn(\012)1374 2739 y Fm(\002)14 b Fo(C)7 b Fs(\(\012\))14 b Fm(\002)g Ff(R)29 b Fp(such)h(that)g Fo(q)2300 2753 y Fk(F)2388 2739 y Fp(is)g(p)-5 b(ositive)31 b(de\014nite.)41 b(Then)30 b(ther)-5 b(e)0 2852 y(exists)37 b(a)g(neighb)-5 b(orho)g(o)g(d)41 b Fo(U)i Fm(\032)32 b Fo(C)1161 2819 y Fn(2)1200 2852 y Fs(\()p Fo(M)5 b(;)15 b Ff(R)1447 2819 y Fk(n)1494 2852 y Fs(\))24 b Fm(\002)f Fo(C)7 b Fs(\(\012\))23 b Fm(\002)g Ff(R)36 b Fp(of)h Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))39 b Fp(such)e(that)h(al)5 b(l)37 b Fs(\()3134 2829 y(~)3113 2852 y Fo(F)13 b(;)24 b Fs(~)-54 b Fo(p)q(;)3313 2828 y Fs(~)3311 2852 y Fo(k)s Fs(\))33 b Fm(2)g Fo(U)0 2965 y Fp(with)214 2942 y Fs(~)194 2965 y Fo(F)38 b Fm(2)25 b(E)424 2979 y Fn(\012)479 2965 y Fp(,)30 b(satis\014es)g(that)h Fo(q)1114 2978 y Fn(~)1099 2995 y Fk(F)1186 2965 y Fp(is)e(p)-5 b(ositive)30 b(de\014nite,)g(and)g (the)g(c)-5 b(onstant)31 b Fo(C)36 b Fp(given)28 b(by)h(L)-5 b(emma)31 b(4.3.9)0 3088 y(is)i(indep)-5 b(endent)34 b(of)f Fs(\()764 3065 y(~)744 3088 y Fo(F)13 b(;)23 b Fs(~)-53 b Fo(p;)944 3064 y Fs(~)941 3088 y Fo(k)t Fs(\))25 b Fm(2)g Fo(U)10 b Fp(.)0 3269 y(Pr)-5 b(o)g(of)p Fs(.)43 b(W)-8 b(e)32 b(write)e Fo(q)709 3283 y Fk(F)798 3269 y Fs(in)f(terms)i(of)f(a)h(con)m(tin)m(uous)g(function)e(de\014ned)h (on)g Fo(J)2648 3236 y Fn(2)2639 3296 y(\012)2695 3269 y Fs(.)41 b(Let)31 b Fo(s)2967 3283 y Fk(p)3032 3269 y Fs(:)26 b Fo(J)3142 3236 y Fn(2)3133 3296 y(\012)3209 3269 y Fm(\002)20 b Fo(T)3366 3236 y Fe(\003)3406 3269 y Fo(M)30 b Fm(\002)0 3382 y Fl(R)78 3349 y Fk(n)151 3382 y Fm(!)25 b Fl(R)31 b Fs(b)s(e)e(giv)m(en)i(b)m(y)179 3616 y Fo(s)222 3630 y Fk(p)261 3616 y Fs(\()p Fo(j)338 3578 y Fn(2)333 3638 y Fk(x)378 3616 y Fo(F)s(;)15 b(\030)t(;)g(X)7 b Fs(\))85 b(=)919 3492 y Fi(Z)969 3698 y Fn(\012)1020 3712 y Fg(F)8 b Fb(\()p Fg(x)p Fb(\))1173 3616 y Fo(h)1225 3578 y Fe(\003)1265 3616 y Fs(\()p Fo(j)1342 3578 y Fn(2)1337 3638 y Fk(x)1382 3616 y Fo(F)13 b Fs(\))1488 3542 y Fi(\000)1530 3616 y Fo(\030)t(;)i Fs([\()p Fo(I)29 b Fm(\000)19 b Fo(S)1889 3630 y Fk(v)1930 3616 y Fs(\()p Fo(j)2007 3578 y Fn(2)2002 3638 y Fk(x)2048 3616 y Fo(F)13 b Fs(\)\))2189 3578 y Fe(\000)p Fn(1)2284 3616 y Fs(])2309 3578 y Fe(\003)2348 3616 y Fo(\030)2392 3542 y Fi(\001)2434 3616 y Fm(h)p Fo(X)r(;)i(v)s Fm(i)2668 3578 y Fn(2)2729 3616 y Fm(\002)1464 3848 y Fo(p)p Fs(\()p Fo(F)e Fs(\()p Fo(x)p Fs(\))21 b(+)f Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2324 3862 y Fk(v)2364 3848 y Fs(\()p Fo(j)2441 3811 y Fn(2)2436 3871 y Fk(x)2482 3848 y Fo(F)13 b Fs(\)\))p Fo(dv)2717 3766 y Fi(p)p 2809 3766 277 4 v 2809 3848 a Fo(h)p Fs(\()p Fo(j)2938 3822 y Fn(2)2933 3871 y Fk(x)2979 3848 y Fo(F)g Fs(\))p Fo(=)3130 3766 y Fi(p)p 3222 3766 170 4 v 3222 3848 a Fo(g)s Fs(\()p Fo(x)p Fs(\))q Fo(:)-3416 b Fs(\(4.57\))0 4045 y(By)31 b(form)m(ula)e(\(4)p Fo(:)p Fs(33\),)k(w)m(e)e(can)g(write)e Fo(q)1338 4059 y Fk(F)1396 4045 y Fs(\()p Fo(X)f Fm(\012)20 b Fo(\030)t(;)15 b(X)28 b Fm(\012)20 b Fo(\030)t Fs(\))31 b(as)821 4242 y Fo(q)862 4256 y Fk(F)920 4242 y Fs(\()p Fo(X)d Fm(\012)20 b Fo(\030)t(;)15 b(X)28 b Fm(\012)20 b Fo(\030)t Fs(\))25 b(=)g Fo(k)s(g)1723 4204 y Fe(\003)1763 4242 y Fs(\()p Fo(\030)t(;)15 b(\030)t Fs(\))p Fm(j)p Fo(X)7 b Fm(j)2093 4204 y Fn(2)2155 4242 y Fm(\000)20 b Fo(s)2289 4256 y Fk(p)2328 4242 y Fs(\()p Fo(j)2405 4204 y Fn(2)2446 4242 y Fo(F)s(;)15 b(\030)t(;)g(X)7 b Fs(\))p Fo(:)0 4438 y Fs(T)-8 b(o)32 b(pro)m(v)m(e)g(the)f(lemma)g(w) m(e)h(sho)m(w)f(that)h(this)e(form)m(ula)h(de\014nes)f(a)i(con)m(tin)m (uous)f(function)f(in)g(b)s(oth)g(v)-5 b(ari-)0 4551 y(ables)30 b Fo(F)43 b Fs(and)30 b Fo(p)p Fs(.)136 4664 y(Consider)44 b(the)i(space)f Fm(C)51 b Fs(of)46 b(con)m(tin)m(uous)f (functions)f Fo(f)59 b Fs(:)51 b Fo(J)2294 4631 y Fn(2)2285 4691 y(\012)2370 4664 y Fm(\002)30 b Fo(T)2537 4631 y Fe(\003)2577 4664 y Fo(M)40 b Fm(\002)30 b Fl(R)2884 4631 y Fk(n)2982 4664 y Fm(!)50 b Fl(R)45 b Fs(with)g(the)0 4777 y(top)s(ology)32 b(of)g(the)f(uniform)f(con)m(v)m(ergence)j(on)f (compact)h(sets.)44 b(Consider)30 b(also)h(the)h(comp)s(osition)e(map)0 4890 y Fo(C)i Fs(:)25 b(\()p Fo(C)254 4857 y Fn(2)294 4890 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)541 4857 y Fk(n)588 4890 y Fs(\))20 b Fm(\\)g(E)772 4904 y Fn(\012)827 4890 y Fs(\))h Fm(\002)f Fo(C)7 b Fs(\(\012\))25 b Fm(!)g(C)35 b Fs(giv)m(en)c(b)m(y)1076 5013 y Fi(\000)1117 5087 y Fo(C)7 b Fs(\()p Fo(F)s(;)15 b(p)p Fs(\))1406 5013 y Fi(\001)1448 5087 y Fs(\()p Fo(j)1525 5049 y Fn(2)1520 5109 y Fk(x)1566 5087 y Fo(F)s(;)g(\030)t(;)g(X)7 b Fs(\))27 b(=)e Fo(s)2034 5101 y Fk(p)2073 5087 y Fs(\()p Fo(j)2150 5049 y Fn(2)2145 5109 y Fk(x)2190 5087 y Fo(F)s(;)15 b(\030)t(;)g(X)7 b Fs(\))p Fo(:)0 5284 y Fs(W)-8 b(e)22 b(will)c(pro)m(v)m(e)j(that)h Fo(C)27 b Fs(is)19 b(con)m(tin)m(uous.)38 b(Assume)20 b(this)f(for)i(no)m(w)f(and)h(supp)s(ose)e(that)i(the)g (lemma)f(is)g(false.)0 5396 y(Then)k(there)h(w)m(ould)e(b)s(e)h (sequences)h Fm(f)p Fs(\()p Fo(F)1381 5410 y Fk(n)1430 5396 y Fo(;)15 b(p)1516 5410 y Fk(n)1563 5396 y Fo(;)g(k)1650 5410 y Fk(n)1697 5396 y Fs(\))p Fm(g)26 b(\032)f Fs(\()p Fo(C)2006 5363 y Fn(2)2046 5396 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2293 5363 y Fk(n)2340 5396 y Fs(\))9 b Fm(\\)g(E)2502 5410 y Fn(\012)2557 5396 y Fs(\))g Fm(\002)g Fo(C)e Fs(\(\012\))i Fm(\002)g Fl(R)p Fo(;)30 b Fm(f)p Fs(\()p Fo(x)3243 5410 y Fk(n)3291 5396 y Fo(;)15 b(\030)3371 5410 y Fk(n)3418 5396 y Fs(\))p Fm(g)26 b(\032)p eop %%Page: 45 53 45 52 bop 3803 100 a Fs(45)299 490 y Fo(T)365 457 y Fe(\003)404 490 y Fo(M)42 b Fs(with)31 b Fo(g)789 457 y Fe(\003)829 490 y Fs(\()p Fo(\030)904 504 y Fk(n)951 490 y Fo(;)15 b(\030)1031 504 y Fk(n)1078 490 y Fs(\))28 b(=)g(1)33 b(and)e Fm(f)p Fo(X)1616 504 y Fk(n)1664 490 y Fm(g)d(\032)g Fo(S)1897 457 y Fk(n)p Fe(\000)p Fn(1)2034 490 y Fs(,)33 b(suc)m(h)e(that)i Fm(f)p Fs(\()p Fo(F)2635 504 y Fk(n)2683 490 y Fo(;)15 b(p)2769 504 y Fk(n)2816 490 y Fo(;)g(k)2903 504 y Fk(n)2951 490 y Fs(\))p Fm(g)33 b Fs(con)m(v)m(erges)h(to)e(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))299 603 y(and)37 b Fo(q)524 617 y Fk(F)569 625 y Fg(n)615 603 y Fs(\()p Fo(X)725 617 y Fk(n)797 603 y Fm(\012)25 b Fo(\030)933 617 y Fk(n)980 603 y Fo(;)15 b(X)1095 617 y Fk(n)1167 603 y Fm(\012)25 b Fo(\030)1303 617 y Fk(n)1350 603 y Fs(\))38 b Fm(\024)e Fo(cg)1615 570 y Fe(\003)1656 603 y Fs(\()p Fo(\030)1731 617 y Fk(n)1778 603 y Fo(;)15 b(\030)1858 617 y Fk(n)1905 603 y Fs(\))p Fm(j)p Fo(X)2040 617 y Fk(n)2088 603 y Fm(j)2113 570 y Fn(2)2153 603 y Fs(,)39 b(with)d Fo(c)i(>)f Fs(0)h(is)f(de\014ned)f(as)i(in)e(Lemma)i(4.3.9.)299 716 y(Since)28 b Fo(M)38 b Fs(is)28 b(compact,)i(there)f(exists)g(\()p Fo(x;)15 b(\030)t Fs(\))26 b Fm(2)f Fo(T)2003 683 y Fe(\003)2042 716 y Fo(M)39 b Fs(with)27 b Fo(g)2420 683 y Fe(\003)2460 716 y Fs(\()p Fo(\030)t(;)15 b(\030)t Fs(\))26 b(=)f(1)k(and)f Fo(X)33 b Fm(2)25 b Fo(S)3284 683 y Fk(n)p Fe(\000)p Fn(1)3450 716 y Fs(suc)m(h)j(that)h(a)299 829 y(subsequence)20 b(of)h Fm(f)p Fs(\()p Fo(x)1029 843 y Fk(n)1077 829 y Fo(;)15 b(\030)1157 843 y Fk(n)1204 829 y Fs(\))p Fm(g)22 b Fs(con)m(v)m(erges)h(to)f(\()p Fo(x;)15 b(\030)t Fs(\),)24 b(and)d(a)g(subsequence)f(of)i Fm(f)p Fo(X)3014 843 y Fk(n)3061 829 y Fm(g)g Fs(con)m(v)m(erges)h(to)f Fo(X)7 b Fs(.)38 b(On)299 941 y(the)27 b(other)h(hand,)f(b)m(y)g(the)g(con)m (tin)m(uit)m(y)g(of)g(the)h(comp)s(osition)e(map)g Fo(C)7 b Fs(,)28 b(the)f(sequence)h Fm(f)p Fo(C)7 b Fs(\()p Fo(F)3458 955 y Fk(n)3505 941 y Fo(;)15 b(p)3591 955 y Fk(n)3639 941 y Fs(\))p Fm(g)26 b(\032)f(C)299 1054 y Fs(con)m(v)m(erges)39 b(uniformly)34 b(on)k(compact)g(sets)g(and)e (therefore)i(it)f(is)f(equicon)m(tin)m(uous)g(on)h(compact)i(sets.)299 1167 y(This)28 b(implies)830 1384 y Fo(q)871 1398 y Fk(F)929 1384 y Fs(\()p Fo(X)g Fm(\012)20 b Fo(\030)t(;)15 b(X)28 b Fm(\012)20 b Fo(\030)t Fs(\))26 b(=)54 b(lim)1637 1438 y Fk(n)p Fe(!1)1836 1384 y Fo(q)1877 1398 y Fk(F)1922 1406 y Fg(n)1968 1384 y Fs(\()p Fo(X)2078 1398 y Fk(n)2146 1384 y Fm(\012)19 b Fo(\030)2276 1398 y Fk(n)2323 1384 y Fo(;)c(X)2438 1398 y Fk(n)2506 1384 y Fm(\012)20 b Fo(\030)2637 1398 y Fk(n)2684 1384 y Fs(\))25 b Fm(\024)g Fo(cg)2925 1346 y Fe(\003)2966 1384 y Fs(\()p Fo(\030)t(;)15 b(\030)t Fs(\))p Fm(j)p Fo(X)7 b Fm(j)3296 1346 y Fn(2)3337 1384 y Fo(:)299 1623 y Fs(By)38 b(Lemma)h(4.3.9,)j(this)37 b(is)g(a)h(con)m(tradiction)g(since)g(w)m(e)h(assumed)e(that)i Fo(q)2953 1637 y Fk(F)3049 1623 y Fs(w)m(as)f(p)s(ositiv)m(e)g (de\014nite.)299 1736 y(This)28 b(pro)m(v)m(es)k(Lemma)e(4.3.10.)435 1853 y(W)-8 b(e)25 b(no)m(w)g(pro)m(v)m(e)f(that)h(the)f(comp)s (osition)f(map)h Fo(C)7 b Fs(\()p Fo(F)s(;)15 b(p)p Fs(\))24 b(is)f(con)m(tin)m(uous.)38 b(Let)25 b(\()p Fo(F)s(;)15 b(p)p Fs(\))26 b Fm(2)f Fs(\()p Fo(C)3525 1820 y Fn(2)3564 1853 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3811 1820 y Fk(n)3858 1853 y Fs(\))299 1966 y Fm(\\E)408 1980 y Fn(\012)463 1966 y Fs(\))24 b Fm(\002)f Fo(C)7 b Fs(\(\012\))35 b(and)g(let)h Fo(K)1255 1980 y Fn(1)1328 1966 y Fm(\032)e Fo(J)1492 1933 y Fn(2)1483 1993 y(\012)1539 1966 y Fo(;)c(K)1671 1980 y Fn(2)1745 1966 y Fm(\032)j Fo(T)1915 1933 y Fe(\003)1955 1966 y Fo(M)5 b(;)30 b(K)2180 1980 y Fn(3)2254 1966 y Fm(\032)j Fl(R)2437 1933 y Fk(n)2519 1966 y Fs(compact)k(subsets)e(suc) m(h)g(that)h(the)g(set)299 2079 y Fm(f)p Fo(j)386 2046 y Fn(2)381 2101 y Fk(x)426 2079 y Fo(F)62 b Fm(2)49 b Fo(J)715 2046 y Fn(2)706 2105 y(\012)810 2079 y Fs(:)h Fo(x)e Fm(2)h Fo(M)10 b Fm(g)49 b(\032)g Fo(K)1484 2093 y Fn(1)1524 2079 y Fs(.)83 b(Let)45 b Fo(\017)k(>)f Fs(0.)84 b(Since)44 b Fo(s)2463 2093 y Fk(p)2546 2079 y Fs(is)g(uniformly)e(con) m(tin)m(uous)i(on)g Fo(K)56 b Fs(=)299 2191 y Fo(K)376 2205 y Fn(1)426 2191 y Fm(\002)10 b Fo(K)584 2205 y Fn(2)634 2191 y Fm(\002)g Fo(K)792 2205 y Fn(3)832 2191 y Fs(,)26 b(there)g(exists)f Fo(\016)k(>)c Fs(0)h(suc)m(h)f(that)h(for)f(all)2261 2168 y(~)2241 2191 y Fo(F)38 b Fm(2)25 b Fo(C)2495 2158 y Fn(2)2534 2191 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2781 2158 y Fk(n)2828 2191 y Fs(\))10 b Fm(\\)g(E)2992 2205 y Fn(\012)3073 2191 y Fs(with)24 b Fm(jj)3346 2168 y Fs(~)3325 2191 y Fo(F)g Fm(\000)10 b Fo(F)j Fm(jj)3609 2211 y Fk(C)3664 2192 y Fb(2)3729 2191 y Fo(<)25 b(\016)s Fs(,)299 2304 y(w)m(e)31 b(ha)m(v)m(e)1362 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Fo(I)28 b Fm(\000)20 b Fo(S)2193 5121 y Fk(v)2234 5107 y Fs(\))2269 5069 y Fe(\000)p Fn(1)2364 5107 y Fs(\()p Fm(\000h)p Fo(dW)2638 5122 y Fe(?)2697 5107 y Fo(;)15 b(v)s Fm(i)2819 5069 y Fn(#)2883 5107 y Fs(\))2918 5033 y Fi(\013)1314 5259 y Fs(=)1468 5185 y Fi(\012)1511 5259 y Fo(V)1564 5277 y Fe(k)1603 5259 y Fo(;)g Fs(\()p Fm(\000h)p Fo(dW)1917 5274 y Fe(?)1977 5259 y Fo(;)g(v)s Fm(i)2099 5221 y Fn(#)2163 5259 y Fs(\))2198 5185 y Fi(\013)2267 5259 y Fs(=)25 b Fm(h\000)p Fo(dW)2602 5274 y Fe(?)2661 5259 y Fs(\()p Fo(V)2749 5277 y Fe(k)2789 5259 y Fs(\))p Fo(;)15 b(v)s Fm(i)1314 5396 y Fs(=)83 b Fm(h)p Fo(v)s(;)15 b Fm(\000)p Fo(dW)1794 5411 y Fe(?)1854 5396 y Fs(\()p Fo(V)1942 5415 y Fe(k)1982 5396 y Fs(\))p Fm(i)p Fo(:)p eop %%Page: 48 56 48 55 bop 0 100 a Fs(48)0 490 y(Substituting)28 b(in)m(to)i(\(4)p Fo(:)p Fs(64\),)528 721 y Fo(H)604 735 y Fk(F)662 721 y Fs(\()p Fo(V)750 740 y Fe(k)790 721 y Fo(;)15 b(W)916 736 y Fe(?)975 721 y Fs(\))83 b(=)1247 597 y Fi(Z)1297 804 y Fn(\012)1348 815 y Fg(F)1403 721 y Fm(h)p Fo(v)s(;)15 b Fm(\000)p Fo(dW)1729 736 y Fe(?)1789 721 y Fs(\()p Fo(V)1877 740 y Fe(k)1917 721 y Fs(\))p Fm(i)p Fo(p)p Fs(\()p Fo(F)34 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2725 735 y Fk(v)2766 721 y Fs(\))p Fo(dv)1093 1007 y Fs(=)1247 852 y Fi(*)1340 1007 y Fm(\000)1431 883 y Fi(Z)1482 1089 y Fn(\012)1533 1100 y Fg(F)1603 1007 y Fo(p)p Fs(\()p Fo(F)33 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2340 1021 y Fk(v)2381 1007 y Fs(\))p Fo(v)e(dv)s(;)d(dW)2745 1022 y Fe(?)2806 1007 y Fs(\()p Fo(V)2894 1026 y Fe(k)2933 1007 y Fs(\))2968 852 y Fi(+)3042 1007 y Fo(:)0 1269 y Fs(By)31 b(de\014nition)d(of)i Fo(\034)690 1283 y Fk(F)749 1269 y Fs(,)1200 1458 y Fo(H)1276 1472 y Fk(F)1335 1458 y Fs(\()p Fo(V)1423 1477 y Fe(k)1462 1458 y Fo(;)15 b(W)1588 1473 y Fe(?)1648 1458 y Fs(\))25 b(=)g Fm(h)p Fo(\034)1879 1472 y Fk(F)1938 1458 y Fo(;)15 b(dW)2111 1473 y Fe(?)2171 1458 y Fs(\()p Fo(V)2259 1477 y Fe(k)2298 1458 y Fs(\))p Fm(i)p Fo(:)0 1648 y Fs(Using)30 b(this)f(in)g(the)h(de\014nition)e(of) j Fo(Q)1275 1662 y Fk(F)1364 1648 y Fs(giv)m(es)508 1837 y Fo(Q)580 1851 y Fk(F)638 1837 y Fs(\()p Fo(V)726 1856 y Fe(k)766 1837 y Fo(;)15 b(W)892 1852 y Fe(?)951 1837 y Fs(\))26 b(=)f Fo(k)s(\016)1201 1800 y Fk(ij)1263 1837 y Fm(h)p Fo(dV)1398 1856 y Fe(k)1438 1837 y Fs(\()p Fo(e)1515 1800 y Fk(M)1515 1860 y(i)1594 1837 y Fs(\))p Fo(;)15 b(dW)1802 1852 y Fe(?)1862 1837 y Fs(\()p Fo(e)1939 1800 y Fk(M)1939 1860 y(j)2019 1837 y Fs(\))p Fm(i)21 b(\000)f(h)p Fo(\034)2276 1851 y Fk(F)2335 1837 y Fo(;)15 b(dW)2508 1852 y Fe(?)2567 1837 y Fs(\()p Fo(V)2655 1856 y Fe(k)2695 1837 y Fs(\))p Fm(i)2765 1755 y(p)p 2842 1755 53 4 v 2842 1837 a Fo(h=)2939 1776 y Fm(p)p 3015 1776 47 4 v 61 x Fo(g)t(:)0 2027 y Fs(By)31 b(\(4)p Fo(:)p Fs(5\),)414 2216 y Fo(Q)486 2230 y Fk(F)545 2216 y Fs(\()p Fo(V)633 2235 y Fe(k)672 2216 y Fo(;)15 b(W)798 2231 y Fe(?)858 2216 y Fs(\))25 b(=)g Fo(k)s(\016)1107 2179 y Fk(ij)1169 2216 y Fm(h)p Fo(dV)1304 2235 y Fe(k)1344 2216 y Fs(\()p Fo(e)1421 2179 y Fk(M)1421 2239 y(i)1501 2216 y Fs(\))p Fo(;)15 b(dW)1709 2231 y Fe(?)1769 2216 y Fs(\()p Fo(e)1846 2179 y Fk(M)1846 2239 y(j)1925 2216 y Fs(\))p Fm(i)21 b(\000)f Fo(k)s(\016)2200 2179 y Fk(ij)2262 2216 y Fm(h)p Fo(B)5 b Fs(\()p Fo(e)2448 2179 y Fk(M)2448 2239 y(i)2527 2216 y Fo(;)15 b(e)2609 2179 y Fk(M)2609 2239 y(j)2689 2216 y Fs(\))p Fo(;)g(dW)2897 2231 y Fe(?)2957 2216 y Fs(\()p Fo(V)3045 2235 y Fe(k)3085 2216 y Fs(\))p Fm(i)p Fo(:)0 2415 y Fs(W)-8 b(e)28 b(write)f(the)g(term)g Fm(h)p Fo(dV)885 2433 y Fe(k)925 2415 y Fs(\()p Fo(e)1002 2382 y Fk(M)1002 2440 y(i)1082 2415 y Fs(\))p Fo(;)15 b(dW)1290 2430 y Fe(?)1350 2415 y Fs(\()p Fo(e)1427 2382 y Fk(M)1427 2440 y(j)1507 2415 y Fs(\))p Fm(i)27 b Fs(separating)g(in)m(to)g (normal)g(and)f(tangen)m(tial)i(comp)s(onen)m(ts)0 2540 y(of)j Fo(dV)204 2558 y Fe(k)243 2540 y Fs(\()p Fo(e)320 2507 y Fk(M)320 2565 y(i)400 2540 y Fs(\))g(and)e Fo(dW)775 2555 y Fe(?)835 2540 y Fs(\()p Fo(e)912 2507 y Fk(M)912 2565 y(j)991 2540 y Fs(\).)41 b(W)-8 b(e)32 b(get)391 2746 y Fo(Q)463 2760 y Fk(F)522 2746 y Fs(\()p Fo(V)610 2764 y Fe(k)649 2746 y Fo(;)15 b(W)775 2761 y Fe(?)835 2746 y Fs(\))83 b(=)g Fo(k)s(\016)1200 2708 y Fk(ij)1261 2672 y Fi(\000)1323 2746 y Fm(\000)20 b(h)p Fo(S)1505 2760 y Fk(W)1572 2772 y Fa(?)1628 2746 y Fo(e)1670 2708 y Fk(M)1670 2768 y(j)1750 2746 y Fo(;)15 b(dV)1890 2764 y Fe(k)1930 2746 y Fs(\()p Fo(e)2007 2708 y Fk(M)2007 2768 y(i)2086 2746 y Fs(\))p Fm(i)21 b Fs(+)f Fm(h)p Fo(B)5 b Fs(\()p Fo(e)2454 2708 y Fk(M)2454 2768 y(i)2534 2746 y Fo(;)15 b(V)2627 2764 y Fe(k)2667 2746 y Fs(\))p Fo(;)g(dW)2875 2761 y Fe(?)2935 2746 y Fs(\()p Fo(e)3012 2708 y Fk(M)3012 2768 y(j)3091 2746 y Fs(\))p Fm(i)3161 2672 y Fi(\001)1107 2897 y Fm(\000)p Fo(k)s(\016)1271 2860 y Fk(ij)1332 2897 y Fm(h)p Fo(B)5 b Fs(\()p Fo(e)1518 2860 y Fk(M)1518 2920 y(i)1598 2897 y Fo(;)15 b(e)1680 2860 y Fk(M)1680 2920 y(j)1759 2897 y Fs(\))p Fo(;)g(dW)1967 2912 y Fe(?)2027 2897 y Fs(\()p Fo(V)2115 2916 y Fe(k)2155 2897 y Fs(\))p Fm(i)p Fo(:)0 3087 y Fs(Using)30 b(\(4)p Fo(:)p Fs(34\))i(giv)m(es)f(\(4)p Fo(:)p Fs(62\).)0 3290 y Fp(Pr)-5 b(o)g(of)34 b(of)f Fs(\(4)p Fo(:)p Fs(63\).)136 3492 y(By)e(\(3)p Fo(:)p Fs(18\),)462 3682 y Fo(H)538 3696 y Fk(F)596 3682 y Fs(\()p Fo(V)684 3697 y Fe(?)744 3682 y Fo(;)15 b(W)870 3697 y Fe(?)929 3682 y Fs(\))308 3862 y(=)462 3738 y Fi(Z)513 3944 y Fn(\012)564 3955 y Fg(F)634 3788 y Fi(\012)677 3862 y Fo(\031)729 3876 y Fk(F)787 3862 y Fo(V)840 3877 y Fe(?)919 3862 y Fm(\000)20 b(h)p Fo(dV)1145 3877 y Fe(?)1205 3862 y Fo(;)15 b(v)s Fm(i)1327 3824 y Fn(#)1391 3862 y Fo(;)g Fs(\()p Fo(I)28 b Fm(\000)20 b Fo(S)1681 3876 y Fk(v)1722 3862 y Fs(\))1757 3824 y Fe(\000)p Fn(1)1851 3862 y Fs(\()p Fo(\031)1938 3876 y Fk(F)1997 3862 y Fo(W)2083 3877 y Fe(?)2162 3862 y Fm(\000)g(h)p Fo(dW)2421 3877 y Fe(?)2481 3862 y Fo(;)15 b(v)s Fm(i)2603 3824 y Fn(#)2667 3862 y Fs(\))2702 3788 y Fi(\013)2765 3862 y Fm(\002)2462 4056 y Fo(p)p Fs(\()p Fo(F)33 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)3199 4070 y Fk(v)3240 4056 y Fs(\))p Fo(dv)308 4236 y Fs(=)462 4112 y Fi(Z)513 4319 y Fn(\012)564 4330 y Fg(F)634 4163 y Fi(\012)677 4236 y Fm(h)p Fo(dV)812 4251 y Fe(?)871 4236 y Fs(\))p Fo(;)15 b(v)s Fm(i)1028 4199 y Fn(#)1093 4236 y Fo(;)g Fs(\()p Fo(I)28 b Fm(\000)20 b Fo(S)1383 4250 y Fk(v)1423 4236 y Fs(\))1458 4199 y Fe(\000)p Fn(1)1553 4236 y Fm(h)p Fo(dW)1721 4251 y Fe(?)1781 4236 y Fo(;)15 b(v)s Fm(i)1903 4199 y Fn(#)1967 4163 y Fi(\013)2010 4236 y Fo(p)p Fs(\()p Fo(F)33 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)19 b Fo(S)2746 4250 y Fk(v)2787 4236 y Fs(\))p Fo(dv)s(:)0 4494 y Fs(W)-8 b(e)32 b(ha)m(v)m(e)f Fm(h)p Fo(dV)501 4509 y Fe(?)561 4494 y Fo(;)15 b(v)s Fm(i)683 4461 y Fn(#)772 4494 y Fs(=)25 b Fm(h)p Fs(\()p Fo(dV)1038 4509 y Fe(?)1098 4494 y Fs(\))1133 4509 y Fe(?)1193 4494 y Fo(;)15 b(v)s Fm(i)1315 4461 y Fn(#)1379 4494 y Fs(.)40 b(Hence,)98 4725 y 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y Fs(\))26 b(=)f Fo(k)s(\016)987 5119 y Fk(ij)1049 5156 y Fm(h)p Fo(dV)1184 5171 y Fe(?)1243 5156 y Fs(\()p Fo(e)1320 5119 y Fk(M)1320 5179 y(i)1400 5156 y Fs(\))p Fo(;)15 b(dW)1608 5171 y Fe(?)1668 5156 y Fs(\()p Fo(e)1745 5119 y Fk(M)1745 5179 y(j)1825 5156 y Fs(\))p Fm(i)365 5348 y(\000)451 5225 y Fi(Z)501 5431 y Fn(\012)552 5442 y Fg(F)623 5275 y Fi(\012)665 5348 y Fm(h)p Fs(\()p Fo(dV)835 5363 y Fe(?)896 5348 y Fs(\))931 5363 y Fe(?)990 5348 y Fo(;)g(v)s Fm(i)1112 5311 y Fn(#)1176 5348 y Fo(;)g Fs(\()p Fo(I)28 b Fm(\000)20 b Fo(S)1466 5362 y Fk(v)1507 5348 y Fs(\))1542 5311 y Fe(\000)p Fn(1)1636 5348 y Fs(\()p Fm(h)p Fs(\()p Fo(dW)1874 5363 y Fe(?)1935 5348 y Fs(\))1970 5363 y Fe(?)2029 5348 y Fo(;)15 b(v)s Fm(i)2151 5311 y Fn(#)2215 5275 y Fi(\013)2258 5348 y Fo(p)p Fs(\()p Fo(F)33 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2995 5362 y Fk(v)3036 5348 y Fs(\))p Fo(dv)3165 5267 y Fm(p)p 3241 5267 53 4 v 81 x Fo(h)q(=)3339 5287 y Fm(p)p 3415 5287 47 4 v 61 x Fo(g)s(:)p eop %%Page: 49 57 49 56 bop 3803 100 a Fs(49)299 490 y(W)-8 b(e)31 b(write)f Fm(h)p Fo(dV)823 505 y Fe(?)883 490 y Fs(\()p Fo(e)960 457 y Fk(M)960 515 y(i)1039 490 y Fs(\))p Fo(;)15 b(dW)1247 505 y Fe(?)1307 490 y Fs(\()p Fo(e)1384 457 y Fk(M)1384 515 y(j)1464 490 y Fs(\))p Fm(i)31 b Fs(separating)f(in)m(to)h(normal)e (and)h(tangen)m(tial)g(comp)s(onen)m(ts)h(of)435 615 y Fo(dV)535 630 y Fe(?)595 615 y Fs(\()p Fo(e)672 582 y Fk(M)672 640 y(i)751 615 y Fs(\))g(and)f Fo(dW)1127 630 y Fe(?)1186 615 y Fs(\()p Fo(e)1263 582 y Fk(M)1263 640 y(j)1343 615 y Fs(\).)41 b(By)30 b(\(2)p Fo(:)p Fs(5\),)593 836 y Fo(Q)665 850 y Fk(F)724 836 y Fs(\()p Fo(V)812 851 y Fe(?)871 836 y Fo(;)15 b(W)997 851 y Fe(?)1056 836 y Fs(\))26 b(=)f Fo(k)s(\016)1306 798 y Fk(ij)1368 836 y Fm(h)p Fo(S)1459 850 y Fk(V)1500 862 y Fa(?)1556 836 y Fo(e)1598 798 y Fk(M)1598 858 y(i)1677 836 y Fo(;)15 b(S)1773 850 y Fk(W)1840 862 y Fa(?)1896 836 y Fo(e)1938 798 y Fk(M)1938 858 y(j)2018 836 y Fm(i)20 b Fs(+)g 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53 4 v 3520 1028 a Fo(h=)3617 967 y Fm(p)p 3694 967 47 4 v 3694 1028 a Fo(g)s(:)299 1284 y Fs(This)28 b(can)j(b)s(e)f(written)f(in)g(terms)i(of)f Fo(q)1619 1298 y Fk(F)1708 1284 y Fs(as)930 1489 y Fo(Q)1002 1503 y Fk(F)1060 1489 y Fs(\()p Fo(V)1148 1504 y Fe(?)1208 1489 y Fo(;)15 b(W)1334 1504 y Fe(?)1393 1489 y Fs(\))26 b(=)f Fo(k)s(\016)1643 1451 y Fk(ij)1704 1489 y Fm(h)p Fo(S)1795 1503 y Fk(V)1836 1515 y Fa(?)1893 1489 y Fo(e)1935 1451 y Fk(M)1935 1511 y(i)2014 1489 y Fo(;)15 b(S)2110 1503 y Fk(W)2177 1515 y Fa(?)2233 1489 y Fo(e)2275 1451 y Fk(M)2275 1511 y(j)2354 1489 y Fm(i)21 b Fs(+)f Fo(q)2542 1503 y Fk(F)2600 1489 y Fs(\()p Fm(r)2711 1451 y Fe(?)2770 1489 y Fo(V)2823 1504 y Fe(?)2882 1489 y Fo(;)15 b Fm(r)2998 1451 y Fe(?)3057 1489 y Fo(W)3143 1504 y Fe(?)3202 1489 y Fs(\))p Fo(:)299 1693 y Fs(Using)29 b(\(4)p Fo(:)p Fs(34\))k(giv)m(es)e(\(4)p Fo(:)p Fs(63\).)p eop %%Page: 50 58 50 57 bop 1583 805 a Fq(Chapter)34 b(5)688 1005 y Fj(ST)-9 b(ABILITY)36 b(OF)i(CRITICAL)e(EMBEDDINGS)136 1284 y Fs(In)k(this)f(c)m(hapter)h(w)m(e)h(are)f(concerned)g(with)f(the)h(b)s (eha)m(vior)f(of)h(the)h(functional)d Fo(E)45 b Fs(near)40 b(a)h(lo)s(cal)0 1396 y(minim)m(um.)49 b(W)-8 b(e)35 b(consider)e(a)h(probabilit)m(y)e(densit)m(y)h Fo(p)1882 1410 y Fn(0)1955 1396 y Fs(and)h(a)g(spring)e(constan)m(t)k Fo(k)2910 1410 y Fn(0)2983 1396 y Fs(suc)m(h)e(that)h(there)0 1509 y(exists)29 b(a)i(lo)s(cal)e(minim)m(um)e Fo(F)1004 1523 y Fn(0)1069 1509 y Fm(2)e(E)1203 1523 y Fn(\012)1288 1509 y Fs(of)30 b Fo(E)5 b Fs(.)40 b(Let)31 b(a)f(pair)f(\()p Fo(p;)15 b(k)s Fs(\))30 b Fp(ne)-5 b(ar)41 b Fs(\()p Fo(p)2483 1523 y Fn(0)2522 1509 y Fo(;)15 b(k)2609 1523 y Fn(0)2650 1509 y Fs(\))30 b(in)e(a)i(suitable)f(top)s(ology)-8 b(.)0 1622 y(W)g(e)32 b(address)f(the)g(question)g(of)g(existence)h (and)f(uniqueness)e(of)i(a)h(critical)e(em)m(b)s(edding)f Fo(F)40 b Fs(:)27 b Fo(M)37 b Fm(!)26 b Fl(R)3548 1589 y Fk(n)0 1735 y Fs(asso)s(ciated)34 b(with)f(\()p Fo(p;)15 b(k)s Fs(\))34 b(suc)m(h)g(that)g(it)f(lo)s(cally)g(minimizes)e Fo(E)5 b Fs(.)51 b(W)-8 b(e)35 b(call)e(this)g(problem)f(the)i Fp(stability)0 1848 y(pr)-5 b(oblem)39 b Fs(for)30 b Fo(E)5 b Fs(.)136 1961 y(In)44 b(the)h(follo)m(wing)e(theorem,)49 b(w)m(e)c(sho)m(w)g(that)g(critical)f(curv)m(es)h(that)g(lo)s(cally)e (minimize)f Fo(E)50 b Fs(are)0 2074 y(preserv)m(ed)30 b(under)f(small)g(p)s(erturbations)f(of)i Fo(p)h Fs(and)e Fo(k)s Fs(.)0 2240 y Fl(Theorem)34 b(5.0.12)47 b Fp(L)-5 b(et)33 b Fo(k)28 b Fm(\025)d Fs(2)p Fo(;)31 b(u)1238 2254 y Fn(0)1303 2240 y Fm(2)25 b Fo(S)1450 2207 y Fn(1)1489 2240 y Fp(,)33 b(and)g(let)g Fo(F)1912 2254 y Fn(0)1978 2240 y Fm(2)25 b(E)2120 2202 y Fk(k)r Fn(+2)2112 2268 y(\012)2285 2240 y Fp(b)-5 b(e)33 b(a)g(critic)-5 b(al)33 b(emb)-5 b(e)g(dding)34 b(of)f Fo(E)38 b Fp(with)0 2353 y(pr)-5 b(ob)g(ability)37 b(density)e Fo(p)799 2367 y Fn(0)873 2353 y Fp(and)h(spring)f(c)-5 b(onstant)37 b Fo(k)1741 2367 y Fn(0)1781 2353 y Fp(.)48 b(If)34 b Fo(P)2012 2367 y 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Fn(0)1236 2691 y Fs(\))25 b Fm(2)g Fo(N)1455 2710 y Fk(F)1500 2719 y Fb(0)1535 2710 y Fn(\()p Fk(u)1603 2719 y Fb(0)1638 2710 y Fn(\))1669 2691 y Fp(,)33 b(such)f(that)i(it)f(lo)-5 b(c)g(al)5 b(ly)34 b(minimizes)f Fo(E)5 b Fp(.)0 2857 y Fs(The)21 b(follo)m(wing)f(examples)h(sho)m(w)g(that)h(the)f(class)h(of)f (critical)f(em)m(b)s(eddings)g(that)i(satisfy)f(the)g(h)m(yp)s(othesis) 0 2970 y(of)31 b(Theorem)f(5.0.12)i(is)e(not)g(empt)m(y)-8 b(.)0 3136 y Fl(Example)34 b(5.0.13)46 b Fs(F)-8 b(or)35 b(a)g(critical)d(em)m(b)s(edding)h Fo(F)1819 3150 y Fn(0)1893 3136 y Fs(in)f(an)i(ann)m(ular)f(distribution)d(giv)m(en)k(b)m(y)g(\(4) p Fo(:)p Fs(16\),)0 3249 y(w)m(e)26 b(ha)m(v)m(e)g(that)g Fo(P)583 3263 y Fk(F)628 3272 y Fb(0)692 3249 y Fs(is)e(a)i(p)s(ositiv) m(e)e(constan)m(t)i(and)f Fo(q)1754 3263 y Fk(F)1799 3272 y Fb(0)1862 3249 y Fs(is)f(p)s(ositiv)m(e)h(de\014nite)f(if)g(the) h(radius)f Fo(r)j Fs(is)e(less)f(than)0 3362 y(1)p Fo(=)p Fs(\(2)p Fo(\031)s(\027)276 3329 y Fn(0)317 3362 y Fs(\),)i(\(Example)e (4.3.5\).)41 b(Our)23 b(stabilit)m(y)g(result)g(implies)f(the)i (existence)h(of)g(a)g(lo)s(cally)d(minimizing)0 3475 y(critical)36 b(curv)m(e)h(for)f(an)m(y)i(probabilit)m(y)c (distribution)f(su\016cien)m(tly)j(close)h(to)g(an)g(ann)m(ular)e (distribution)0 3588 y(and)30 b(for)g(an)m(y)h(spring)d(constan)m(t)k (close)e(to)i Fo(k)1504 3602 y Fn(0)1543 3588 y Fs(.)0 3774 y Fl(Example)i(5.0.14)46 b Fp(T)-7 b(or)i(al)49 b(distribution.)87 b Fs(Consider)44 b(a)i(constan)m(t)i(probabilit)m(y) 43 b(distribution)g(sup-)0 3887 y(p)s(orted)29 b(on)h(a)h(solid)d (torus)i(with)f(inner)f(radius)g Fo(R)j Fs(and)f(transv)m(erse)g (radius)e Fo(r)s Fs(.)40 b(If)30 b(the)g(spring)e(constan)m(t)0 4000 y(satis\014es)1597 4211 y Fo(k)g(>)1783 4150 y Fs(2)p 1778 4190 56 4 v 1778 4273 a Fo(\031)1867 4150 y(r)1911 4117 y Fn(2)p 1853 4190 110 4 v 1853 4273 a Fo(R)1923 4247 y Fn(2)1973 4211 y Fo(;)0 4410 y Fs(then)43 b(an)m(y)h(critical)e (circle)h(concen)m(tric)h(to)g(the)g(torus)f(of)g(radius)f Fo(R)30 b Fm(\000)e Fo(r)50 b(<)c(a)h(<)g(R)q Fs(,)g(satis\014es)c(the) 0 4523 y(h)m(yp)s(othesis)29 b(of)i(Theorem)g(5.0.12.)44 b(T)-8 b(o)31 b(c)m(hec)m(k)h(this,)e(w)m(e)h(\014rst)f(write)g(do)m (wn)g(the)h(equation)g(of)g(a)g(critical)0 4636 y(curv)m(e)g(for)f (this)f(probabilit)m(y)f(distribution.)37 b(W)-8 b(e)32 b(consider)d(the)h(em)m(b)s(edding)954 4814 y Fo(F)13 b Fs(\()p Fo(u)p Fs(\))26 b(=)f Fo(a)p Fs(\(cos)q(\()p Fo(u)p Fs(\))p Fo(;)15 b Fs(sin\()p Fo(u)p Fs(\))p Fo(;)g Fs(0\))p Fo(;)199 b(u)26 b Fm(2)e Fs([0)p Fo(;)15 b Fs(2)p Fo(\031)s Fs(])p Fo(:)0 4992 y Fs(Denote)37 b(b)m(y)d Fl(N)h Fs(the)h(unit)d(normal)h(v)m(ector)j(\014eld)c(along)j Fo(F)13 b Fs(\()p Fo(S)2129 4959 y Fn(1)2168 4992 y Fs(\),)37 b(and)e(b)m(y)g Fl(B)g Fs(the)g(binormal)e(to)j Fo(F)13 b Fs(\()p Fo(S)3520 4959 y Fn(1)3559 4992 y Fs(\))0 5105 y(orien)m(ted)32 b(so)g(that)h(\()p Fl(T)p Fo(;)15 b Fl(N)p Fo(;)g Fl(B)p Fs(\))33 b(is)e(consisten)m(t)h(with)f(the)h (standard)f(orien)m(tation)h(of)g Fl(R)2989 5072 y Fn(3)3029 5105 y Fs(.)45 b(W)-8 b(e)33 b(ha)m(v)m(e)h(the)0 5218 y(iden)m(tities)543 5396 y Fm(h)p Fl(T)651 5359 y Fe(0)674 5396 y Fo(=)p Fm(j)p Fo(F)815 5359 y Fe(0)840 5396 y Fm(j)p Fo(;)15 b Fl(N)p Fm(i)26 b Fs(=)f Fm(\000)p Fs(1)p Fo(=a;)1575 5315 y Fm(p)p 1651 5315 53 4 v 81 x Fo(h)q(=)1749 5335 y Fm(p)p 1825 5335 47 4 v 61 x Fo(g)k Fs(=)c Fo(a;)197 b Fs(and)181 b Fo(F)2662 5359 y Fe(00)2730 5396 y Fs(=)25 b Fm(\000)p Fo(a)p Fl(N)p Fo(:)p eop %%Page: 51 59 51 58 bop 3803 100 a Fs(51)299 490 y(T)-8 b(o)29 b(compute)f Fo(\034)837 504 y Fk(F)896 490 y Fs(,)h(w)m(e)g(note)g(that)g(for)f (this)f(t)m(yp)s(e)i(of)f(distribution)d(the)k(disks)e(\012)3032 504 y Fk(F)r(;u)3171 490 y Fs(are)i(standard)e(disks)299 603 y(of)f(radius)e Fo(r)29 b Fs(p)s(erp)s(endicular)22 b(to)k(the)h(circle)e Fo(F)13 b Fs(\()p Fo(S)1968 570 y Fn(1)2008 603 y Fs(\).)39 b(Then)25 b(the)i(transv)m(erse)f(in)m (tegration)g(along)g(\012)3681 617 y Fk(F)r(;u)3818 603 y Fs(in)299 716 y(the)k(de\014nition)d(of)j Fo(\034)1001 730 y Fk(F)1089 716 y Fs(can)g(b)s(e)f(computed)g(b)m(y)h(making)f(a)h (c)m(hange)h(of)f(v)-5 b(ariables)28 b(to)j(p)s(olar)d(co)s(ordinates) 299 829 y(cen)m(tered)j(at)g Fo(F)13 b Fs(\()p Fo(u)p Fs(\))26 b Fm(2)f Fs(\012)1142 843 y Fk(F)r(;u)1253 829 y Fs(.)41 b(W)-8 b(e)31 b(write)f Fo(v)f Fm(2)24 b Fs(\012)1932 843 y Fk(F)r(;u)2074 829 y Fs(as)1504 1040 y Fo(v)k Fs(=)d Fo(\032)15 b Fs(cos)h Fo(\022)s Fl(N)p Fs(\()p Fo(u)p Fs(\))k(+)g Fo(\032)15 b Fs(sin)f Fo(\022)s Fl(B)p Fs(\()p Fo(u)p Fs(\))p Fo(:)299 1251 y Fs(W)-8 b(e)31 b(ha)m(v)m(e)530 1481 y Fo(\034)570 1495 y Fk(F)628 1481 y Fs(\()p Fo(u)p Fs(\))84 b(=)f Fm(\000)1074 1358 y Fi(Z)1124 1564 y Fn(\012)1175 1575 y Fg(F)r(;u)1294 1481 y Fo(p)p Fs(\()p Fo(F)13 b Fs(\()p Fo(u)p Fs(\))21 b(+)f Fo(v)s Fs(\)\(1)i Fm(\000)e(h)p Fl(T)2063 1444 y Fe(0)2086 1481 y Fo(=)p Fm(j)p Fo(F)2227 1444 y Fe(0)2251 1481 y Fm(j)p Fo(;)15 b(v)s Fm(i)p Fs(\))p Fo(v)20 b(dv)834 1777 y Fs(=)83 b Fm(\000)1074 1653 y Fi(Z)1164 1679 y Fn(2)p Fk(\031)1124 1859 y Fn(0)1262 1653 y Fi(Z)1353 1679 y Fk(\032)p Fn(\()p Fk(\022)r Fn(\))1312 1859 y(0)1498 1777 y Fo(p)p Fs(\()p Fo(\032;)15 b(\022)s Fs(\))1747 1621 y Fi( )1819 1777 y Fs(1)21 b(+)1985 1715 y Fo(\032)15 b Fs(cos)i Fo(\022)p 1985 1756 245 4 v 2084 1839 a(a)2240 1621 y Fi(!)2312 1777 y Fs(\()p Fo(\032)e Fs(cos)h Fo(\022)s Fl(N)p Fs(\()p Fo(u)p Fs(\))k(+)g Fo(\032)15 b Fs(sin)f Fo(\022)s Fl(B)p Fs(\()p Fo(u)p Fs(\)\))p Fo(\032)h(d\032)g(d\022)s(:)299 2065 y Fs(Note)36 b(that)f(in)e(general)h Fo(F)13 b Fs(\()p Fo(u)p Fs(\))35 b(ma)m(y)g(not)g(b)s(e)f(a)g(the)h(cen)m(ter)g(of)g(the)g(disk)e(\012) 2917 2079 y Fk(F)r(;u)3027 2065 y Fs(.)53 b(F)-8 b(or)35 b(this)f(reason,)h(the)299 2178 y(upp)s(er)h(limit)f(in)h(the)i(in)m (tegration)g(with)e(resp)s(ect)i(to)g Fo(\032)g Fs(dep)s(ends)e(on)h Fo(\022)s Fs(.)62 b(An)37 b(elemen)m(tary)h(geometry)299 2291 y(argumen)m(t)31 b(sho)m(ws)f(that)h(suc)m(h)f(dep)s(endence)f(is) h(giv)m(en)g(b)m(y)1212 2537 y Fo(\032)p Fs(\()p Fo(\022)s Fs(\))c(=)e(\()p Fo(R)e Fm(\000)e Fo(a)p Fs(\))15 b(cos)h Fo(\022)23 b Fs(+)2104 2423 y Fi(q)p 2195 2423 760 4 v 114 x Fo(r)2239 2510 y Fn(2)2298 2537 y Fm(\000)d Fs(\()p Fo(R)i Fm(\000)e Fo(a)p Fs(\))2689 2510 y Fn(2)2744 2537 y Fs(sin)2855 2502 y Fn(2)2910 2537 y Fo(\022)r(:)299 2757 y Fs(Since)40 b(the)h(probabilit)m(y)e(densit)m(y)i(is)f(a)h (constan)m(t)i(giv)m(en)e(b)m(y)g Fo(p)i Fs(=)g(1)p Fo(=v)s(ol)r Fs(\(\012\),)j(and)41 b Fo(\032)p Fs(\()p Fo(\022)s Fs(\))g(is)f(an)h (ev)m(en)299 2870 y(function)f(of)h Fo(\022)s Fs(,)i(then)d(the)i(o)s (ccurrence)f(of)g(the)g(o)s(dd)f(function)g(sin)13 b Fo(\022)43 b Fs(as)f(a)f(factor)h(m)m(ultiplying)37 b(the)299 2983 y(binormal)28 b Fl(B)p Fs(\()p Fo(u)p Fs(\))j(mak)m(es)h(the)e(in) m(tegral)g(in)m(v)m(olving)f(the)i(term)f Fl(B)p Fs(\()p Fo(u)p Fs(\))i(v)-5 b(anish.)39 b(Therefore,)30 b(w)m(e)h(get)876 3272 y Fo(\034)916 3286 y Fk(F)975 3272 y Fs(\()p Fo(u)p Fs(\))26 b(=)f Fm(\000)1405 3210 y Fs(1)p 1300 3251 257 4 v 1300 3334 a Fo(v)s(ol)r Fs(\(\012\))1581 3148 y Fi(Z)1672 3174 y Fn(2)p Fk(\031)1632 3354 y Fn(0)1770 3148 y Fi(Z)1861 3174 y Fk(\032)p Fn(\()p Fk(\022)r Fn(\))1820 3354 y(0)2006 3116 y Fi( )2078 3272 y Fs(1)20 b(+)2244 3210 y Fo(\032)15 b Fs(cos)h Fo(\022)p 2244 3251 245 4 v 2342 3334 a(a)2498 3116 y Fi(!)2570 3272 y Fo(\032)2617 3234 y Fn(2)2672 3272 y Fs(cos)g Fo(\022)h(d\032)e(d\022)j Fl(N)p Fs(\()p Fo(u)p Fs(\))p Fo(:)299 3577 y Fs(W)-8 b(e)28 b(no)m(w)e(write)g(the)h (equation)f(of)h(a)g(critical)f(em)m(b)s(edding)f(\(4)p Fo(:)p Fs(5\))j(m)m(ultiplying)23 b Fo(\034)3048 3591 y Fk(F)3133 3577 y Fs(b)m(y)3255 3499 y Fm(p)p 3331 3499 53 4 v 78 x Fo(h)q(=)3429 3520 y Fm(p)p 3505 3520 47 4 v 57 x Fo(g)29 b Fs(=)c Fo(a)h Fs(and)299 3690 y(adding)j Fo(k)s Fs(\001)723 3704 y Fk(g)763 3690 y Fo(F)38 b Fs(=)25 b Fm(\000)p Fo(k)s(a)p Fl(N)p Fs(,)459 3982 y Fo(\034)499 3996 y Fk(F)567 3843 y Fm(p)p 643 3843 53 4 v 77 x Fo(h)p 567 3961 129 4 v 570 3988 a Fm(p)p 646 3988 47 4 v 56 x Fo(g)725 3982 y Fs(+)20 b Fo(k)s Fs(\001)942 3996 y Fk(g)982 3982 y Fo(F)39 b Fs(=)25 b Fm(\000)1361 3920 y Fs(1)p 1256 3961 257 4 v 1256 4044 a Fo(v)s(ol)r Fs(\(\012\))1537 3858 y Fi(Z)1628 3884 y Fn(2)p Fk(\031)1588 4064 y Fn(0)1725 3858 y Fi(Z)1816 3884 y Fk(\032)p Fn(\()p Fk(\022)r Fn(\))1776 4064 y(0)1962 3826 y Fi( )2034 3982 y Fs(1)20 b(+)2200 3920 y Fo(\032)15 b Fs(cos)h Fo(\022)p 2200 3961 245 4 v 2298 4044 a(a)2454 3826 y Fi(!)2526 3982 y Fo(\032)2573 3944 y Fn(2)2628 3982 y Fs(cos)g Fo(\022)h(d\032)e(d\022)j(a)p Fl(N)p Fs(\()p Fo(u)p Fs(\))j Fm(\000)f Fo(k)s(a)p Fl(N)p Fs(\()p Fo(u)p Fs(\))p Fo(:)299 4265 y Fs(This)28 b(expression)i(v)-5 b(anishes)29 b(if)g(and)h(only)f(if)993 4549 y Fm(\000)1180 4487 y Fs(1)p 1074 4528 257 4 v 1074 4611 a Fo(v)s(ol)r Fs(\(\012\))1356 4425 y Fi(Z)1447 4451 y Fn(2)p Fk(\031)1406 4631 y Fn(0)1544 4425 y Fi(Z)1635 4451 y Fk(\032)p Fn(\()p Fk(\022)r Fn(\))1595 4631 y(0)1780 4393 y Fi( )1852 4549 y Fs(1)21 b(+)2019 4487 y Fo(\032)15 b Fs(cos)h Fo(\022)p 2019 4528 245 4 v 2117 4611 a(a)2273 4393 y Fi(!)2345 4549 y Fo(\032)2392 4511 y Fn(2)2447 4549 y Fs(cos)g Fo(\022)h(d\032)e(d\022)23 b Fm(\000)d Fo(k)28 b Fs(=)d(0)p Fo(:)-2899 b Fs(\(5.1\))299 4832 y(Recall)32 b(that)i(for)f(a)g (radially)e(symmetric)h(distribution)d(relativ)m(e)k(to)h(a)f(curv)m(e) h(\000,)f(suc)m(h)g(\000)f(is)g(a)i(critical)299 4945 y(em)m(b)s(edding)27 b(of)j Fo(E)35 b Fs(\(Example)29 b(4.2.3\).)42 b(Since)29 b(the)g(toral)h(distribution)25 b(is)k(symmetric)f(with)h(resp)s(ect)g(to)299 5058 y(the)i(inner)e (circle)g(of)i(radius)e Fo(a)d Fs(=)f Fo(R)q Fs(,)31 b(equation)f(\(5)p Fo(:)p Fs(1\))j(is)c(satis\014ed)h(b)m(y)g(this)g (circle)g(for)g(the)h(v)-5 b(alue)30 b Fo(k)3750 5072 y Fn(0)3821 5058 y Fs(of)299 5171 y(the)h(spring)f(constan)m(t)j(giv)m (en)e(in)f(Example)h(4.2.3.)45 b(Since)31 b(form)m(ula)f(\(5)p Fo(:)p Fs(1\))j(de\014nes)e(a)h(smo)s(oth)f(function)299 5284 y(of)g(\()p Fo(a;)15 b(k)s Fs(\))32 b(for)e Fo(R)21 b Fm(\000)f Fo(r)28 b(<)e(a)g(<)f(R)q Fs(,)31 b(an)f(sho)m(ws)g (function)g(theorem)h(argumen)m(t)g(for)f(real)h(v)-5 b(alued)29 b(functions)299 5396 y(implies)21 b(that)j(there)g(exists)f Fo(\017)i(>)g Fs(0)f(suc)m(h)g(that)g(for)g Fo(k)k Fm(2)d Fs(\()p Fo(k)2253 5410 y Fn(0)2300 5396 y Fm(\000)7 b Fo(\017;)15 b(k)2502 5410 y Fn(0)2548 5396 y Fs(+)7 b Fo(\017)p Fs(\))24 b(there)g(exists)f(a)h(concen)m(tric)g(circle)p eop %%Page: 52 60 52 59 bop 0 100 a Fs(52)0 490 y(to)33 b(the)g(torus)f(with)f(radius)f (in)h(a)i(neigh)m(b)s(orho)s(o)s(d)d(of)j Fo(R)g Fs(suc)m(h)f(that)h (it)f(de\014nes)f(a)i(critical)e(em)m(b)s(edding.)0 603 y(Moreo)m(v)m(er,)i(By)d(Example)g(4.3.3,)i(the)f(symmetric)f(tensor)g Fo(q)2101 617 y Fk(F)2190 603 y Fs(is)f(p)s(ositiv)m(e)g(de\014nite)h (if)f(and)h(only)f(if)982 816 y Fo(k)s Fm(j)p Fo(X)7 b Fm(j)1164 778 y Fn(2)1225 816 y Fm(\000)20 b Fs(\003)1379 830 y Fn(2)1418 816 y Fs(\()p Fo(X)r(;)15 b(X)7 b Fs(\))p 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3079 y Fk(n)2093 3112 y Fs(\))p Fo(;)g Fs(\000)2225 3079 y Fk(k)2268 3112 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)2516 3079 y Fk(n)2563 3112 y Fs(\)\).)51 b(This)31 b(is)i(a)h(second)g(order)f(partial)299 3225 y(di\013eren)m(tial)i(op)s(erator)i(since)f(it)h(is)f(the)h (linearization)e(of)h(the)i(op)s(erator)f Fo(L)p Fs(\()p Fm(\001)p Fo(;)15 b(p;)g(k)s Fs(\))38 b(at)f Fo(F)13 b Fs(.)61 b(T)-8 b(o)37 b(pro)m(v)m(e)299 3338 y(that)31 b Fo(\016)s(L)p Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))32 b(is)d(elliptic,)f(w)m(e)j(compute)f(its)g(principal)d(sym)m(b)s(ol)i Fo(\033)f Fs(:)e Fo(T)2870 3305 y Fe(\003)2909 3338 y Fo(M)35 b Fm(!)25 b Fo(H)7 b(om)p Fs(\()p Fl(R)3469 3305 y Fk(n)3516 3338 y Fo(;)15 b Fl(R)3635 3305 y Fk(n)3682 3338 y Fs(\))30 b(and)299 3451 y(sho)m(w)g(that)h(this)e(is)h(injectiv) m(e)g(for)g(all)f(\()p Fo(x;)15 b(\030)t Fs(\))27 b Fm(2)d Fo(T)2000 3418 y Fe(\003)2040 3451 y Fo(M)10 b Fs(.)435 3564 y(Let)44 b Fo(V)5 b(;)15 b(W)60 b Fm(2)46 b Fo(C)1034 3531 y Fe(1)1108 3564 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)1355 3531 y Fk(n)1402 3564 y Fs(\))44 b(and)e(denote)i(b)m(y)f Fm(h)15 b Fo(;)g Fm(i)2240 3578 y Fn(0)2324 3564 y Fs(the)43 b(scalar)h(pro)s(duct)e(in)g(\000)3295 3531 y Fn(0)3334 3564 y Fs(\()p Fo(M)5 b(;)15 b Fl(R)3581 3531 y Fk(n)3628 3564 y Fs(\).)79 b(W)-8 b(e)299 3677 y(compute)685 3876 y Fi(\012)728 3950 y Fo(\016)s(L)p Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))22 b Fm(\001)f Fo(V)5 b(;)15 b(W)1405 3876 y Fi(\013)1448 3986 y Fn(0)1570 3950 y Fs(=)85 b(lim)1724 4008 y Fk(t)p Fe(!)p Fn(0)1870 3795 y Fi(*)1953 3889 y Fo(L)p Fs(\()p Fo(F)34 b Fs(+)20 b Fo(tV)5 b(;)15 b(p;)g(k)s Fs(\))21 b Fm(\000)f Fo(L)p Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))p 1953 3929 1064 4 v 2469 4012 a Fo(t)3027 3950 y(;)g(W)3166 3795 y Fi(+)3239 4068 y Fn(0)1570 4250 y Fs(=)85 b(lim)1724 4308 y Fk(t)p Fe(!)p Fn(0)1880 4105 y Fi(\012)1923 4178 y Fo(L)p Fs(\()p Fo(F)34 b Fs(+)20 b Fo(tV)5 b(;)15 b(p;)g(k)s Fs(\))p Fo(;)g(W)2644 4105 y Fi(\013)2687 4214 y Fn(0)2747 4178 y Fm(\000)2838 4105 y Fi(\012)2881 4178 y Fo(L)p Fs(\()p Fo(F)s(;)g(p;)g(k)s Fs(\))p Fo(;)g(W)3389 4105 y Fi(\013)3433 4214 y Fn(0)p 1880 4229 1593 4 v 2660 4312 a Fo(t)3482 4250 y(:)-3208 b Fs(\(5.3\))299 4498 y(By)30 b(Theorem)g(4.1.1,)j(the)d(map)g Fo(E)g Fs(:)c Fm(E)1630 4460 y Fk(k)r Fn(+2)1622 4526 y(\012)1783 4498 y Fm(\002)19 b Fo(C)1945 4465 y Fk(k)r Fn(+2)2077 4498 y Fs(\(\012\))i Fm(\002)e Fl(R)26 b Fm(!)f Fl(R)30 b Fs(is)g Fo(C)2816 4465 y Fn(2)2855 4498 y Fs(.)40 b(Denote)32 b(b)m(y)e Fo(\016)s(E)36 b Fs(its)30 b(partial)299 4611 y(deriv)-5 b(ativ)m(e)38 b(with)e(resp)s(ect)i(to)h(the)g(\014rst) e(argumen)m(t.)64 b(F)-8 b(rom)39 b(the)f(\014rst)g(v)-5 b(ariation)37 b(form)m(ula)g(\(4)p Fo(:)p Fs(4\),)42 b(w)m(e)299 4724 y(write)29 b Fo(\016)s(E)37 b Fs(in)29 b(terms)h(of)h Fo(L)f Fs(as)1207 4851 y Fi(\012)1250 4925 y Fo(L)p Fs(\()p Fo(F)j Fs(+)20 b Fo(tV)5 b(;)15 b(p;)g(k)s Fs(\))p Fo(;)g(W)1970 4851 y Fi(\013)2014 4961 y Fn(0)2079 4925 y Fs(=)25 b Fo(\016)s(E)5 b Fs(\()p Fo(F)35 b Fs(+)19 b Fo(tV)5 b(;)15 b(p;)g(k)s Fs(\))22 b Fm(\001)e Fo(W)m(:)299 5126 y Fs(Using)29 b(this)h(in)f(\(5)p Fo(:)p Fs(3\))j(giv)m(es)788 5301 y Fi(\012)831 5375 y Fo(\016)s(L)p Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))23 b Fm(\001)d Fo(V)5 b(;)15 b(W)1508 5301 y Fi(\013)1551 5411 y Fn(0)1616 5375 y Fs(=)27 b(lim)1712 5433 y Fk(t)p Fe(!)p Fn(0)1868 5314 y Fo(\016)s(E)5 b Fs(\()p Fo(F)35 b Fs(+)20 b Fo(tV)5 b(;)15 b(p;)g(k)s Fs(\))21 b Fm(\001)g Fo(W)32 b Fm(\000)20 b Fo(\016)s(E)5 b Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))23 b Fm(\001)d Fo(W)p 1868 5354 1501 4 v 2602 5437 a(t)3379 5375 y(:)p eop %%Page: 54 62 54 61 bop 0 100 a Fs(54)0 490 y(This)29 b(limit)f(exists)i(and)g(is)f (giv)m(en)h(b)m(y)h(the)f(second)h(v)-5 b(ariation)29 b(form)m(ula)917 618 y Fi(\012)960 691 y Fo(\016)s(L)p Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))22 b Fm(\001)f Fo(V)5 b(;)15 b(W)1637 618 y Fi(\013)1680 727 y Fn(0)1744 691 y Fs(=)25 b Fo(\016)1883 654 y Fn(2)1923 691 y Fo(E)5 b Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))22 b Fm(\001)f Fs(\()p Fo(V)5 b(;)15 b(W)e Fs(\))p Fo(;)-2662 b Fs(\(5.4\))0 893 y(whic)m(h)29 b(b)m(y)h(Theorem)g(4.3.1,)j(it)d(can)g(b)s(e)g (written)g(as)523 1058 y Fi(\012)566 1131 y Fo(\016)s(L)p Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))23 b Fm(\001)d Fo(V)5 b(;)15 b(W)1243 1058 y Fi(\013)1286 1167 y Fn(0)1351 1131 y Fs(=)1447 1007 y Fi(Z)1497 1214 y Fk(M)1591 1131 y Fo(P)1649 1145 y Fk(F)1708 1131 y Fm(h)p Fo(V)5 b(;)15 b(W)e Fm(i)p Fo(dv)s(ol)2140 1146 y Fk(h)2207 1131 y Fs(+)2298 1007 y Fi(Z)2348 1214 y Fk(M)2442 1131 y Fo(Q)2514 1145 y Fk(F)2573 1131 y Fs(\()p Fo(V)5 b(;)15 b(W)e Fs(\))p Fo(dv)s(ol)3005 1145 y Fk(g)3046 1131 y Fo(:)-3071 b Fs(\(5.5\))0 1379 y(T)-8 b(o)37 b(compute)g(the)f(principal)d(sym)m(b)s (ol)i(of)i Fo(\016)s(L)p Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))38 b(w)m(e)f(in)m(tegrate)h(b)m(y)e(parts)g(\(5)p Fo(:)p Fs(5\),)k(and)c(consider)0 1492 y(the)31 b(highest)e(order)h (deriv)-5 b(ativ)m(e)30 b(terms.)41 b(By)30 b(de\014nition)e(of)j Fo(Q)2136 1506 y Fk(F)2194 1492 y Fs(,)g(these)g(are)g(giv)m(en)f(b)m (y)452 1730 y Fo(k)518 1606 y Fi(Z)568 1812 y Fk(M)662 1730 y Fo(\016)705 1692 y Fk(ij)767 1730 y Fm(h)p Fo(dV)21 b Fs(\()p Fo(e)1000 1692 y Fk(M)1000 1752 y(i)1079 1730 y Fs(\))p Fo(;)15 b(dW)e Fs(\()p Fo(e)1377 1692 y Fk(M)1377 1752 y(j)1457 1730 y Fs(\))p Fm(i)p Fo(dv)s(ol)1692 1744 y Fk(g)452 1969 y Fm(\000)538 1846 y Fi(Z)589 2052 y Fk(M)683 1846 y Fi(Z)733 2052 y Fn(\012)784 2063 y Fg(F)854 1896 y Fi(\012)897 1969 y Fm(h)p Fo(dV)5 b(;)15 b(v)s Fm(i)1159 1932 y Fn(#)1224 1969 y Fo(;)g Fs(\()p Fo(I)28 b Fm(\000)20 b Fo(S)1514 1983 y Fk(v)1555 1969 y Fs(\))1590 1932 y Fe(\000)p Fn(1)1684 1969 y Fs(\()p Fm(h)p Fo(dW)m(;)15 b(v)s Fm(i)2006 1932 y Fn(#)2072 1969 y Fs(\))2107 1896 y Fi(\013)2150 1969 y Fo(p)p Fs(\()p Fo(F)33 b Fs(+)20 b Fo(v)s Fs(\))15 b(det)q(\()p Fo(I)28 b Fm(\000)20 b Fo(S)2887 1983 y Fk(v)2928 1969 y Fs(\))p Fo(dv)f(dv)s(ol)3238 1984 y Fk(h)3283 1969 y Fo(:)-3308 b Fs(\(5.6\))0 2228 y(Consider)32 b(a)i(lo)s(cal)f(orthonormal)g(frame)h Fm(f)p Fo(e)1550 2242 y Fk(i)1579 2228 y Fm(g)d(\032)g Fo(T)13 b(M)43 b Fs(with)32 b(resp)s(ect)i(to)h(the)f(metric)f Fo(h)p Fs(,)i(and)e(denote)0 2341 y(b)m(y)j Fm(r)208 2355 y Fk(i)271 2341 y Fs(the)g(co)m(v)-5 b(arian)m(t)37 b(deriv)-5 b(ativ)m(e)35 b(along)h Fo(e)1542 2355 y Fk(i)1571 2341 y Fs(.)57 b(W)-8 b(e)36 b(use)g(the)g(iden)m(tit)m(y)2480 2267 y Fi(\012)2523 2341 y Fm(h)p Fo(dV)5 b(;)15 b(v)s Fm(i)2785 2308 y Fn(#)2850 2341 y Fo(;)g(e)2932 2355 y Fk(i)2961 2267 y Fi(\013)3038 2341 y Fs(=)34 b Fm(h)p Fo(v)s(;)15 b Fm(r)3341 2355 y Fk(i)3370 2341 y Fo(V)20 b Fm(i)36 b Fs(to)0 2454 y(write)30 b(\(5)p Fo(:)p Fs(6\))i(as)390 2697 y(\(5)p Fo(:)p Fs(6\))c(=)d Fo(k)764 2573 y Fi(Z)814 2780 y Fk(M)908 2697 y Fo(\016)951 2660 y Fk(ij)1013 2697 y Fm(h)p Fo(dV)c Fs(\()p Fo(e)1246 2660 y Fk(M)1246 2720 y(i)1325 2697 y Fs(\))p Fo(;)15 b(dW)e Fs(\()p Fo(e)1623 2660 y Fk(M)1623 2720 y(j)1704 2697 y Fs(\))p Fm(i)p Fo(dv)s(ol)1939 2711 y Fk(g)481 2937 y 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4601 53 4 v 77 x Fo(h)p 3284 4719 129 4 v 3287 4745 a Fm(p)p 3363 4745 47 4 v 57 x Fo(g)3422 4740 y(:)0 4993 y Fs(By)31 b(de\014nition)d(of)i Fo(q)691 5007 y Fk(F)749 4993 y Fs(,)h(it)f(satis\014es)1102 5195 y Fm(h)p Fo(\033)s Fs(\()p Fo(x;)15 b(\030)t Fs(\))p Fo(X)r(;)g(Y)23 b Fm(i)i Fs(=)g Fo(q)1788 5209 y Fk(F)1847 5195 y Fs(\()p Fo(X)j Fm(\012)20 b Fo(\030)t(;)15 b(Y)40 b Fm(\012)20 b Fo(\030)t Fs(\))2423 5209 y Fk(x)2467 5195 y Fo(:)0 5396 y Fs(If)30 b Fo(q)132 5410 y Fk(F)220 5396 y Fs(is)g(p)s(ositiv)m(e)f(de\014nite,)h(then)g Fo(\033)s Fs(\()p Fo(x;)15 b(\030)t Fs(\))31 b(is)f(injectiv)m(e.)40 b(This)29 b(pro)m(v)m(es)i(that)g Fo(\016)s(L)p Fs(\()p Fo(F)s(;)15 b(p;)g(k)s Fs(\))32 b(is)e(elliptic.)p eop %%Page: 55 63 55 62 bop 3803 100 a Fs(55)299 490 y Fc(5.2)105 b(Pro)s(of)36 b(of)f(Theorem)f(5.0.12)299 693 y Fs(W)-8 b(e)28 b(apply)d(the)i (Implicit)d(F)-8 b(unction)27 b(Theorem)f(as)h(in)e([12)q(],)j(p.164.) 41 b(Fix)26 b Fo(u)2837 707 y Fn(0)2902 693 y Fm(2)f Fo(S)3049 660 y 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Fs(\000)2640 895 y Fk(k)r Fn(+2)2640 959 y(0)2772 933 y Fs(\()p Fo(S)2868 900 y Fn(1)2907 933 y Fo(;)15 b Fl(R)3026 900 y Fk(n)3073 933 y Fs(\)\))t Fm(\002)t Fo(C)3294 900 y Fk(k)r Fn(+2)3426 933 y Fs(\(\012\))t Fm(\002)t Fl(R)o Fs(.)38 b(By)299 1046 y(Theorem)26 b(5.1.1,)j(this)c(is)g(a)i Fo(C)1322 1013 y Fn(1)1387 1046 y Fs(map.)39 b(W)-8 b(e)27 b(sho)m(w)f(that)h(its)f(deriv)-5 b(ativ)m(e)26 b Fo(\016)s(L)2830 1060 y Fn(0)2896 1046 y Fs(at)h(\()p Fo(F)3096 1060 y Fn(0)3136 1046 y Fo(;)15 b(p)3222 1060 y Fn(0)3262 1046 y Fo(;)g(k)3349 1060 y Fn(0)3389 1046 y Fs(\))26 b(is)g(injectiv)m(e.) 299 1159 y(Set)g Fo(W)38 b Fs(=)25 b Fo(V)46 b Fs(in)25 b(\(5)p Fo(:)p Fs(4\),)k(and)c(write)g Fo(V)46 b Fs(=)25 b Fo(f)10 b Fl(T)1834 1173 y Fn(0)1884 1159 y Fs(+)i Fo(V)2020 1174 y Fe(?)2078 1159 y Fs(,)27 b(where)f Fo(f)35 b Fs(=)24 b Fm(h)p Fl(T)2672 1173 y Fn(0)2712 1159 y Fo(;)15 b(V)21 b Fm(i)26 b Fs(and)f Fo(V)3112 1174 y Fe(?)3197 1159 y Fs(=)g Fo(V)32 b Fm(\000)12 b(h)p Fl(T)3569 1173 y Fn(0)3607 1159 y Fo(;)j(V)21 b Fm(i)p Fl(T)3829 1173 y Fn(0)3868 1159 y Fs(.)299 1272 y(W)-8 b(e)31 b(use)g(the)f (second)h(v)-5 b(ariation)29 b(form)m(ula)h(\(4)p Fo(:)p Fs(38\))i(to)f(write)f(\(5)p Fo(:)p Fs(4\))i(as)772 1476 y Fm(h)p Fo(\016)s(L)912 1490 y Fn(0)953 1476 y Fs(\()p Fo(F)1046 1490 y Fn(0)1086 1476 y Fo(;)15 b(p)1172 1490 y Fn(0)1212 1476 y Fo(;)g(k)1299 1490 y Fn(0)1339 1476 y Fs(\))20 b Fm(\001)h Fo(V)5 b(;)15 b(V)20 b Fm(i)1646 1490 y Fn(0)619 1656 y Fs(=)772 1532 y Fi(Z)823 1739 y Fk(S)870 1720 y Fb(1)923 1582 y Fi(\010)976 1656 y Fo(P)1034 1670 y Fk(F)1079 1679 y Fb(0)1118 1656 y Fm(j)p Fo(V)1196 1671 y Fe(?)1255 1656 y Fm(j)1280 1619 y Fn(2)1320 1656 y Fm(j)p Fo(F)1416 1619 y Fe(0)1403 1679 y Fn(0)1443 1656 y Fm(j)h Fs(+)f Fo(k)1630 1582 y Fi(\000)1671 1656 y Fo(f)1726 1619 y Fe(0)1749 1656 y Fo(=)p Fm(j)p Fo(F)1890 1619 y Fe(0)1877 1679 y Fn(0)1918 1656 y Fm(j)g(\000)g(h)p Fl(T)2162 1619 y Fe(0)2162 1679 y Fn(0)2202 1656 y Fo(=)p Fm(j)p Fo(F)2343 1619 y Fe(0)2330 1679 y Fn(0)2370 1656 y Fm(j)p Fo(;)15 b(V)2488 1671 y Fe(?)2548 1656 y Fm(i)2583 1582 y Fi(\001)2625 1605 y Fn(2)2685 1656 y Fs(+)k Fo(q)2816 1670 y Fk(F)2875 1656 y Fs(\()p Fm(r)2986 1619 y Fe(?)3045 1656 y Fo(V)3098 1671 y Fe(?)3157 1656 y Fo(;)c Fm(r)3273 1619 y Fe(?)3332 1656 y Fo(V)3385 1671 y Fe(?)3444 1656 y Fs(\))3479 1582 y Fi(\011)3532 1656 y Fo(du;)-3357 b Fs(\(5.7\))299 1901 y(whic)m(h)35 b(is)g(nonnegativ)m(e)i(since)f Fo(q)1443 1915 y Fk(F)1488 1924 y Fb(0)1562 1901 y Fs(is)g(p)s(ositiv)m (e)f(de\014nite.)57 b(It)37 b(v)-5 b(anishes)35 b(if)g(and)h(only)f(if) g Fo(V)3465 1916 y Fe(?)3560 1901 y Fs(=)g(0)h(and)299 2014 y Fo(f)354 1981 y Fe(0)404 2014 y Fs(=)26 b(0.)45 b(Since)30 b Fo(F)912 2028 y Fn(0)973 2014 y Fs(+)21 b Fo(V)47 b Fm(2)27 b Fs(\000)1310 1976 y Fk(k)r Fn(+2)1310 2040 y(0)1442 2014 y Fs(\()p Fo(S)1538 1981 y Fn(1)1578 2014 y Fo(;)15 b Fl(R)1697 1981 y Fk(n)1744 2014 y Fs(\),)32 b(then)f Fo(V)21 b Fs(\()p Fo(u)2205 2028 y Fn(0)2244 2014 y Fs(\))28 b Fm(2)f Fo(N)2468 2033 y Fk(F)2513 2042 y Fb(0)2547 2033 y Fn(\()p Fk(u)2615 2042 y Fb(0)2650 2033 y Fn(\))2713 2014 y Fs(whic)m(h)j(implies)f(that)j Fo(f)10 b Fs(\()p Fo(u)3624 2028 y Fn(0)3663 2014 y Fs(\))27 b(=)g(0.)299 2127 y(Therefore)36 b Fo(f)43 b Fs(=)34 b(0)j(iden)m(tically)-8 b(.)55 b(This)34 b(implies)g(that)i Fo(\016)s(L)2310 2141 y Fn(0)2350 2127 y Fs(\()p Fo(F)2443 2141 y Fn(0)2483 2127 y Fo(;)15 b(p)2569 2141 y Fn(0)2609 2127 y Fo(;)g(k)2696 2141 y Fn(0)2736 2127 y Fs(\))36 b(is)f(injectiv)m(e.)57 b(Theorem)36 b(5.1.2)299 2240 y(implies)26 b(that)j Fo(\016)s(L)906 2254 y Fn(0)947 2240 y Fs(\()p Fo(F)1040 2254 y Fn(0)1080 2240 y Fo(;)15 b(p)1166 2254 y Fn(0)1205 2240 y Fo(;)g(k)1292 2254 y Fn(0)1333 2240 y Fs(\))29 b(is)f(elliptic)e(and,)j(in)e(particular,)h (its)g(sp)s(ectrum)g(is)g(discrete.)40 b(Therefore)299 2353 y Fo(\016)s(L)404 2367 y Fn(0)444 2353 y Fs(\()p Fo(F)537 2367 y Fn(0)577 2353 y Fo(;)15 b(p)663 2367 y Fn(0)703 2353 y Fo(;)g(k)790 2367 y Fn(0)830 2353 y Fs(\))23 b(is)e(in)m(v)m(ertible.)36 b(By)23 b(the)f(Implicit)e(F)-8 b(unction)22 b(Theorem,)i(there)f(exists)e(a)i(unique)d(con)m(tin-)299 2466 y(uous)k(implicit)d(op)s(erator)k Fo(G)p Fs(\()p Fo(p;)15 b(k)s Fs(\))26 b(de\014ned)e(in)f(a)i(neigh)m(b)s(orho)s(o)s (d)d(of)j(\()p Fo(p)2719 2480 y Fn(0)2758 2466 y Fo(;)15 b(k)2845 2480 y Fn(0)2886 2466 y Fs(\))25 b Fm(2)g Fo(C)3104 2433 y Fk(k)r Fn(+2)3236 2466 y Fs(\(\012\))9 b Fm(\002)g Fl(R)25 b Fs(and)f(tak-)299 2579 y(ing)f(v)-5 b(alues)23 b(in)g(a)h(neigh)m(b)s(orho)s(o)s(d)d(of)k Fo(F)1597 2593 y Fn(0)1662 2579 y Fm(2)g Fs(\000)1805 2540 y Fk(k)r Fn(+2)1805 2605 y(0)1937 2579 y Fs(\()p Fo(S)2033 2546 y Fn(1)2073 2579 y Fo(;)15 b Fl(R)2192 2546 y Fk(n)2239 2579 y Fs(\))24 b(suc)m(h)g(that)g(it)f(satis\014es)h Fo(L)3162 2593 y Fn(0)3201 2579 y Fs(\()p Fo(G)p Fs(\()p Fo(p;)15 b(k)s Fs(\))p Fo(;)g(p;)g(k)s Fs(\))28 b(=)d(0)299 2692 y(and)30 b Fo(G)p Fs(\()p Fo(p)628 2706 y Fn(0)668 2692 y Fo(;)15 b(k)755 2706 y Fn(0)795 2692 y Fs(\))25 b(=)g Fo(F)1009 2706 y Fn(0)1049 2692 y Fs(.)41 b(Then)29 b Fo(F)39 b Fs(=)25 b Fo(G)p Fs(\()p Fo(p;)15 b(k)s Fs(\))31 b(is)f(a)h(critical)e(em)m(b)s(edding.)435 2805 y(T)-8 b(o)32 b(pro)m(v)m(e)g(that)f(it)g(lo)s(cally)e(minimizes)g Fo(E)36 b Fs(w)m(e)31 b(apply)f(part)h(\(i\))g(of)g(Theorem)g(4.3.1;)i (i.e.,)f(w)m(e)f(v)m(erify)299 2917 y(that)26 b Fo(P)549 2931 y Fk(F)633 2917 y Fo(>)f Fs(0)h(and)e(that)i Fo(q)1204 2931 y Fk(F)1287 2917 y Fs(is)f(p)s(ositiv)m(e)f(de\014nite.)38 b(This)23 b(follo)m(ws)i(b)m(y)g(Lemmas)g(4.3.10)i(and)e(4.3.8,)j (since)299 3030 y Fo(P)357 3044 y Fk(F)402 3053 y Fb(0)466 3030 y Fo(>)d Fs(0)31 b(and)e(that)i Fo(q)1052 3044 y Fk(F)1097 3053 y Fb(0)1166 3030 y Fs(is)e(p)s(ositiv)m(e)g(de\014nite.) p eop %%Page: 56 64 56 63 bop 1583 805 a Fq(Chapter)34 b(6)1385 1005 y Fj(CONCLUSIONS)136 1284 y Fs(In)c(this)f(thesis)h(w)m(e)h(studied)d(the)j(v)-5 b(ariational)29 b(prop)s(erties)g(of)h(the)h(functional)1562 1488 y Fo(E)g Fs(:)25 b Fm(E)1758 1502 y Fn(\012)1838 1488 y Fm(!)h Fl(R)0 1692 y Fs(on)35 b(the)h(class)g(of)f(\012-regular) g(em)m(b)s(eddings)f(of)i(a)f(manifold)f Fo(M)46 b Fs(in)m(to)35 b(a)h(closed)f(region)h(\012)d Fm(\032)h Fl(R)3340 1659 y Fk(n)3387 1692 y Fs(.)57 b(W)-8 b(e)0 1805 y(c)m(haracterized)43 b(the)f(critical)f(em)m(b)s(eddings)f(of)i(this)f(functional)f(b)m(y)i (the)g(Euler-Lagrange)g(equation)0 1918 y(\(4)p Fo(:)p Fs(5\).)47 b(In)31 b(the)h(sp)s(ecial)f(case)i(when)e Fo(M)42 b Fs(is)31 b(di\013eomorphic)f(to)i(the)h(circle,)e(w)m(e)i(ga) m(v)m(e)h(conditions)c(under)0 2031 y(whic)m(h)k(critical)g(em)m(b)s (eddings)f(are)j(lo)s(cal)e(minima)g(\(Theorem)h(4.3.1\).)57 b(Finally)-8 b(,)35 b(w)m(e)h(ga)m(v)m(e)h(conditions)0 2144 y(under)i(whic)m(h)g(critical)g(curv)m(es)i(dep)s(end)d(smo)s (othly)i(on)g(b)s(oth)g(the)g(probabilit)m(y)e(densit)m(y)i Fo(p)g Fs(and)g(the)0 2257 y(spring)28 b(constan)m(t)k Fo(k)s Fs(,)f(while)d(remaining)h(lo)s(cal)g(minima)g(of)h Fo(E)36 b Fs(\(Theorem)30 b(5.0.12\).)136 2469 y(A)45 b(n)m(um)m(b)s(er)f(of)h(op)s(en)g(questions)f(remain.)84 b(Tw)m(o)45 b(imp)s(ortan)m(t)f(sp)s(ecial)g(cases)h(of)h(the)f (manifold)0 2582 y(\014tting)27 b(problem)f(are)j(the)e(curv)m(e)i(and) e(surface)g(\014tting)h(problems.)38 b(The)27 b(problem)f(of)i(global)f (existence)0 2695 y(of)34 b(regular)f(lo)s(cal)g(minima)e(remains)h(op) s(en,)j(ev)m(en)f(in)e(the)i(relativ)m(ely)f(simple)f(case)i(of)g(curv) m(es.)51 b(In)33 b(the)0 2808 y(case)26 b(of)g(surfaces,)g(w)m(e)g(ha)m (v)m(e)g(not)g(y)m(et)g(succeeded)g(in)e(\014nding)f(an)i(easily)f(v)m (eri\014able)g(su\016cien)m(t)h(condition)0 2921 y(for)30 b(a)h(critical)e(surface)h(to)i(b)s(e)d(a)i(lo)s(cal)f(minim)m(um)d(of) k Fo(E)5 b Fs(.)p eop %%Page: 57 65 57 64 bop 1647 807 a Fj(BIBLIOGRAPHY)672 1087 y Fs([1])47 b(J.)36 b(Ban\014eld)g(and)h(A.E.)g(Raftery)-8 b(.)39 b(Ice)e(\015o)s(e)g(iden)m(ti\014cation)f(in)f(satellite)i(images)g (using)814 1200 y(mathematical)i(morphology)f(and)h(clustering)e(ab)s (out)i(principal)c(curv)m(es.)40 b Fp(Journal)h(of)814 1313 y(the)33 b(A)n(meric)-5 b(an)32 b(Statistic)-5 b(al)35 b(Asso)-5 b(ciation)p Fs(,)32 b Fl(87)f Fs(\(1992\),)i(7-16.)672 1551 y([2])47 b(J.F.)26 b(Brinkley)-8 b(.)26 b(Kno)m(wledge-driv)m(en)f (ultrasonic)g(three-dimensional)f(organ)i(mo)s(deling.)814 1664 y Fp(IEEE)31 b(T)-7 b(r)i(ans.)34 b(Pat.)f(A)n(nal.)f(Mach.)h (Intel)5 b(l.)p Fs(,)31 b Fl(7)f Fs(\(July)f(1985\),)k(431-441.)672 1901 y([3])47 b(M.)30 b(doCarmo.)h Fp(R)n(iemannian)j(Ge)-5 b(ometry)p Fs(.)32 b(Birkh\177)-45 b(auser,)29 b(Boston,)j(1992.)672 2139 y([4])47 b(T.)28 b(Duc)m(hamp)h(and)f(W.)h(Stuetzle.)g(Extremal)g (Prop)s(erties)e(of)i(Principal)d(Curv)m(es)i(in)f(the)814 2252 y(Plane.)j Fp(The)i(A)n(nnals)h(of)g(Statistics)p Fs(,)f Fl(24)f Fs(\(1996\),)i(1511-1520.)672 2490 y([5])47 b(J.)40 b(Eells)g(and)g(J.H.)i(Sampson.)e(Harmonic)g(Mappings)g(on)h (Riemannian)e(Manifolds.)814 2603 y Fp(A)n(mer.)32 b(J.)g(Math)p Fs(,)f Fl(86)g Fs(\(1964\),)i(109-160.)672 2841 y([6])47 b(L.)24 b(F)-8 b(ang)25 b(and)e(D.C.)i(Gossard.)f(Reconstruction)g(of)g (smo)s(oth)g(parametric)f(surfaces)h(from)814 2954 y(unorganized)32 b(data)j(p)s(oin)m(ts.)e(In)g(J.)h(W)-8 b(arren,)35 b(editor.)f Fp(Curves)h(and)i(Surfac)-5 b(es)37 b(in)e(Com-)814 3066 y(puter)e(Vision)f(and)i(Gr)-5 b(aphics)35 b(3.)d(SPIE)p Fs(,)f Fl(1830)g Fs(\(1992\),)i(226-236.)672 3304 y([7])47 b(A.)24 b(Gosh)m(tasb)m(y)-8 b(.)26 b(Surface)e(reconstruction)g(from)g (scattered)h(measuremen)m(ts.)g(In)e(J.)i(W)-8 b(ar-)814 3417 y(ren,)34 b(editor,)h Fp(Curves)h(and)h(Surfac)-5 b(es)36 b(in)g(Computer)i(Vision)e(and)h(Gr)-5 b(aphics)38 b(3.)e(SPIE)p Fs(,)814 3530 y Fl(1830)31 b Fs(\(1992\),)i(247-256.)672 3768 y([8])47 b(T.J.)36 b(Hastie)i(and)e(W.)h(Stuetzle.)h(Principal)c (Curv)m(es.)i Fp(J.)i(A)n(mer.)g(Statist.)i(Asso)-5 b(c.)p Fs(,)39 b Fl(84)814 3881 y Fs(\(1989\),)33 b(502-516.)672 4119 y([9])47 b(F.)31 b(Helein.)f(Harmonic)g(Di\013eomorphisms)f(b)s (et)m(w)m(een)j(Riemannian)d(Manifolds.)g(V)-8 b(aria-)814 4231 y(tional)20 b(metho)s(ds)h(\(P)m(aris,)i(1988\),)i(309-318.)g Fp(Pr)-5 b(o)g(gr)g(ess)26 b(in)e(Nonline)-5 b(ar)26 b(Di\013er)-5 b(ential)24 b(Equa-)814 4344 y(tions)33 b(Applic)-5 b(ations,)34 b(4)p Fs(.)d(Birkh\177)-45 b(auser)29 b(Boston,)j(Boston)f(MA,)g(1990.)627 4582 y([10])47 b(H.)35 b(Hopp)s(e,)g(T.)g(DeRose,)i(T.)e(Duc)m(hamp,)h(J.)f(McDonald,)h(and)f (W.)g(Stuetzle.)g(Surface)814 4695 y(reconstruction)23 b(from)g(unorganized)g(p)s(oin)m(ts.)f Fp(Computer)28 b(Gr)-5 b(aphics)p Fs(,)28 b Fl(26)c Fs(\(1992\),)j(71-78.)627 4933 y([11])47 b(H.)41 b(Hopp)s(e,)j(T.)e(DeRose,)k(T.)41 b(Duc)m(hamp,)k(J.)c(McDonald,)k(and)c(W.)h(Stuetzle.)g(Mesh)814 5046 y(optimization.)29 b Fp(Computer)34 b(Gr)-5 b(aphics)p Fs(,)33 b Fl(27)e Fs(\(1993\),)i(19-26.)627 5284 y([12])47 b(V.)h(Khatsk)m(evic)m(h)h(and)f(D.)h(Shoiykhet.)f Fp(Di\013er)-5 b(entiable)49 b(Op)-5 b(er)g(ators)52 b(and)e(Nonline)-5 b(ar)814 5396 y(Equations)p Fs(.)31 b(Birkh\177)-45 b(auser,)29 b(Berlin,)g(1994.)p eop %%Page: 58 66 58 65 bop 0 100 a Fs(58)328 490 y([13])47 b(T.)20 b(Kludas.)f(Three)h (dimensional)e(surface)i(reconstruction)h(with)e(the)i(Zeiss)e (photogram-)515 603 y(metric)46 b(industrial)d(measuremen)m(t)k(system) g(InduSURF)e(Digital.)i Fp(A)n(utomotive)g(A)n(u-)515 716 y(tomation)p Fs(.)32 b(Cro)m(ydon)e(UK,)h(1995.)328 945 y([14])47 b(C.T.)23 b(Lim,)g(G.M.)i(T)-8 b(urkiyy)m(ah,)24 b(M.A.)g(Gan)m(ter,)i(D.W.)e(Storti.)f Fp(Implicit)k(R)-5 b(e)g(c)g(onstruction)515 1058 y(of)35 b(Solids)h(fr)-5 b(om)35 b(Cloud)h(Point)f(Sets)p Fs(.)e(A)m(CM)g(3rd)f(symp)s(osium)f (in)g(solid)g(mo)s(deling)f(and)515 1171 y(applications,)e(Salt)i(Lak)m (e)i(Cit)m(y)-8 b(,)30 b(Utah,)i(1995.)328 1400 y([15])47 b(R.W.)32 b(Martin,)f(G.)h(Bashein,)f(M.L.)h(Nessly)f(and)g(F.)h (Sheehan.)f(Methology)h(for)f(three-)515 1513 y(dimensional)c (reconstruction)i(of)h(the)g(left)g(v)m(en)m(tricle)g(from)f (transesophageal)h(ec)m(ho)s(car-)515 1626 y(diograms.)g Fp(Ultr)-5 b(asound)34 b(in)f(Me)-5 b(d.)32 b(and)i(Biol.)p Fs(,)d Fl(19)g Fs(\(1993\),)i(27-38.)328 1855 y([16])47 b(S.)29 b(Muraki.)g(V)-8 b(olumetric)30 b(shap)s(e)f(description)e(of)j (range)h(data)f(using)e(\\blobb)m(y)h(mo)s(del".)515 1968 y Fp(Computer)34 b(Gr)-5 b(aphics)p Fs(,)32 b Fl(25)f Fs(\(July)e(1991\),)k(227-235.)328 2197 y([17])47 b(R.B.)30 b(Sc)m(h)m(udy)f(and)g(D.H.)h(Ballard.)f(Mo)s(del)g(detection)h(of)f (cardiac)h(c)m(ham)m(b)s(ers)f(in)f(ultra-)515 2310 y(sound)22 b(images.)i(T)-8 b(ec)m(hnical)23 b(Rep)s(ort)h(12,)i(Computer)d (Science)g(Departmen)m(t,)k(Univ)m(ersit)m(y)515 2423 y(of)j(Ro)s(c)m(hester,)i(1978.)328 2652 y([18])47 b(R.B.)29 b(Sc)m(h)m(udy)f(and)g(D.H.)h(Ballard.)f(T)-8 b(o)m(w)m(ards)29 b(an)f(anatomical)h(mo)s(del)f(of)g(the)h(hart)f(mo-)515 2765 y(tion)g(as)h(seen)g(in)f(4-d)h(cardiac)g(ultrasound)e(data.)j Fp(Pr)-5 b(o)g(c)g(e)g(e)g(dings)33 b(of)f(the)g(6th)g(Confer)-5 b(enc)g(e)515 2878 y(on)34 b(Computer)i(Applic)-5 b(ations)36 b(in)e(R)-5 b(adiolo)g(gy)36 b(and)f(Computer-A)n(ide)-5 b(d)36 b(A)n(nalysis)e(of)g(R)-5 b(a-)515 2990 y(diolo)g(gic)g(al)35 b(Images)p Fs(,)c(1979.)328 3220 y([19])47 b(F.M.)38 b(Sheehan,)g(M.P)-8 b(.)38 b(F)-8 b(eneley)g(,)40 b(N.P)-8 b(.)38 b(DeBruijin,)f(J.S.)f(Rankin,)i(J.W.)f(Da)m(vis,)j(E.L.)515 3333 y(Bolson,)30 b(P)-8 b(.S.)31 b(Glass,)f(and)g(F.M.)h(Clemen)m(ts.) f(Quan)m(titativ)m(e)h(analysis)e(of)h(regional)g(w)m(all)515 3445 y(thic)m(k)m(ening)21 b(b)m(y)h(transesophageal)h(ec)m(ho)s (cardiograph)m(y)-8 b(.)23 b Fp(Journal)j(of)f(Thor)-5 b(acic)27 b(and)e(Car-)515 3558 y(diovascular)34 b(Sur)-5 b(gery)p Fs(,)31 b Fl(103)g Fs(\(1992\),)j(347-354.)328 3787 y([20])47 b(F.)36 b(Solina)f(and)g(R.)h(Ba)5 b(jcsy)-8 b(.)38 b(Reco)m(v)m(ery)g(of)e(parametric)g(mo)s(dels)e(form)i(range)g (images:)515 3900 y(The)20 b(case)i(for)f(sup)s(erquadratics)d(with)i (global)g(deformations.)h Fp(IEEE)i(T)-7 b(r)i(ans.)25 b(Pat.)f(A)n(nal.)515 4013 y(Mach.)32 b(Intel)5 b(l.)p Fs(,)31 b Fl(12)g Fs(\(F)-8 b(ebruary)30 b(1990\),)j(131-147.)328 4242 y([21])47 b(S.)39 b(Sun)m(thank)-5 b(ar.)39 b(Three)h(dimensional) d(solid)h(ob)5 b(ject)41 b(laser)e(scanner.)h Fp(Pr)-5 b(o)g(c)g(e)g(e)g(dings)44 b(of)515 4355 y(the)c(SPIE)p Fs(.)d(The)g(In)m(ternational)h(So)s(ciet)m(y)g(for)f(Optical)g (Engineering.)f(V)-8 b(ol.)39 b(1826,)j(pp.)515 4468 y(495-510,)33 b(1992.)328 4697 y([22])47 b(H.)39 b(Urak)-5 b(a)m(w)m(a.)40 b Fp(Calculus)h(of)f(V)-7 b(ariations)42 b(and)f(Harmonic)g(Maps)p Fs(.)e(American)f(Mathe-)515 4810 y(matical)30 b(So)s(ciet)m(y)-8 b(,)31 b(Pro)m(vidence,)g(1993.) 328 5039 y([23])47 b(B.C.)33 b(V)-8 b(em)m(uri,)34 b(A.)f(Mitic)m(he)g (and)g(J.K.)f(Aggarw)m(al.)j(Curv)-5 b(ature-based)32 b(represen)m(tation)515 5152 y(of)e(ob)5 b(jects)31 b(form)f(range)h (data.)g Fp(Image)i(and)h(Vision)e(Computing)p Fs(,)g Fl(4)f Fs(\(1986\),)i(107-114.)p eop %%Page: 59 67 59 66 bop 1956 805 a Fj(VIT)-9 b(A)435 1084 y Fs(Jos)m(\023)-43 b(e)30 b(Luis)c(Mart)-10 b(\023)-35 b(\020nez)29 b(Morales)f(w)m(as)h (b)s(orn)d(on)i(Octob)s(er)g(22,)i(1965)g(in)d(Mexico)i(Cit)m(y)-8 b(.)40 b(He)28 b(attended)299 1197 y(the)42 b(Univ)m(ersidad)d (Nacional)j(Aut\023)-45 b(onoma)42 b(de)g(M)m(\023)-43 b(exico)43 b(where)f(he)f(got)i(his)d(degree)j(of)f(Matem\023)-45 b(atico)299 1310 y(in)34 b(Septem)m(b)s(er)h(1991.)58 b(He)36 b(en)m(tered)g(the)g(Univ)m(ersit)m(y)f(of)h(W)-8 b(ashington)35 b(in)g(1992)i(and)e(completed)g(the)299 1423 y(requiremen)m(ts)29 b(for)h(the)h(Do)s(ctor)h(of)e(Philosoph)m(y) e(degree)k(in)d(1998.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF