Content-Type: multipart/mixed; boundary="-------------0110300253636" This is a multi-part message in MIME format. ---------------0110300253636 Content-Type: text/plain; name="01-400.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-400.keywords" minimal geodesics, differentiability, Holder continuity, low regularity ---------------0110300253636 Content-Type: application/postscript; name="geo.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="geo.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: geo.dvi %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Roman Times-Italic Times-Bold Symbol %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips geo %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.10.30:0929 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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3068 y Fq(2)1301 3098 y FE(.)22 b(W)m(ith)g Fu(a)p Fk(\()p Fu(r)r Fk(\))e(=)f Fj(\000)9 b(j)o Fu(r)r Fj(j)1927 3060 y Fm(k)1968 3098 y Ft(=)g FE(log)f Fj(j)p Fu(r)r Fj(j)p FE(,)23 b(both)e(the)h(change)e (of)i(coordinates)e(and)h(the)h(ne)n(w)520 3220 y(metric)i(is)d Fu(C)890 3190 y Fm(k)922 3220 y FE(.)k(The)f(minimal)g(geodesic)f Fv(g)i FE(is)g(mapped)e(to)f Fu(t)27 b Fj(!)18 b Fu(t)6 b Fk(\()p FE(cos)j Fu(a)p Fk(\()n Fu(t)d Fk(\))p Ft(;)j FE(sin)p Fk(\()p Fu(a)p Fk(\()n Fu(t)d Fk(\)\)\))25 b FE(which)f(is)h(precisely)516 3342 y Fu(C)573 3312 y Fm(k)q Fi(+)p Fq(1)704 3342 y FE(at)c(the)f(origin.)686 3464 y(The)h(remainder)f(of)i(this)g(paper)f(is)h(de)n(v)n(oted)f(to)h (the)g(study)f(of)h(the)f(re)o(gularity)f(of)i(a)g(minimal)f(geodesic) 520 3586 y Fv(g)26 b FE(parametrized)e(by)h(arc-length)f(if)i Fu(g)g FE(is)h(a)22 b Fu(C)1855 3556 y Fq(0)p Fl(;)p Fr(a)1946 3586 y FE(-Riemannian)i(for)h(0)c Fj(\024)g Fv(a)h Fj(\024)f FE(1.)26 b(Since)g(it,)g(a)g(priori,)f(is)i(not)520 3708 y(clear)19 b(that)g Fv(g)h FE(is)g(piece)n(wise)15 b Fu(C)1372 3678 y Fq(1)1407 3708 y FE(,)20 b(we)g(will)g(use)f(follo)n (wing)f(de\002nition)g(of)h(a)g(minimal)g(geodesic)f(parametrized)520 3830 y(by)k(arc-length)g(that)h(tak)o(es)g(into)g(account)f(that)h(a)h (minimal)e(geodesic)h(realizes)g(the)g(distance)g(between)f(tw)o(o)520 3952 y(points.)d(\(See)h(also)h([3)o(])g(for)e(other)g(de\002nitions)h (of)g(geodesics)f(when)h(the)g(metric)g(has)g(lo)n(w)h(re)o(gularity)-5 b(.\))688 4074 y(D)8 b FF(E)g(F)g(I)g(N)g(I)g(T)g(I)g(O)g(N)31 b FE(1.3)r(.)45 b(A)26 b(minimal)f(geodesic)g(parametrized)f(by)h (arc-length)e(is)k(a)f(continuous)e(curv)o(e)520 4196 y Fv(g)p Fk(\()n Fu(t)6 b Fk(\))p FE(,)20 b Fu(a)e Fj(\024)e Fu(t)24 b Fj(\024)18 b Fu(b)p FE(,)i(such)g(that)1443 4372 y Fu(d)t Fk(\()p Fv(g)p Fk(\()p Fu(s)p Fk(\))p Ft(;)9 b Fv(g)p Fk(\()n Fu(t)d Fk(\)\))20 b(=)c Fu(t)h Fj(\000)12 b Fu(s)p Ft(;)173 b Fu(t)2309 4384 y Fq(0)2362 4372 y Fj(\024)18 b Fu(s)h Fj(\024)d Fu(t)24 b Fj(\024)16 b Fu(t)2728 4384 y Fq(1)2762 4372 y Ft(:)686 4548 y FE(Then)j(the)h (result)h(of)e(this)i(paper)e(can)h(be)g(summarized)f(in)688 4670 y(T)10 b FF(H)g(E)g(O)g(R)g(E)g(M)21 b FE(1.4)r(.)41 b Fu(If)20 b(g)f(is)h(a)15 b(C)1591 4639 y Fq(0)p Fl(;)p Fr(a)1682 4670 y Fu(-Riemannian)i(metric)j(on)f(a)g(smooth)g(manifold)f (and)g Fv(g)i Fu(is)g(a)g(minimal)520 4792 y(g)o(eodesic)f(par)o (ametrized)g(by)h(ar)m(c-length,)f(then)g(for)i Fv(a)e Fk(=)e FE(1)p Fu(,)j Fv(g)f Fj(2)c Fu(C)2473 4761 y Fq(1)p Fl(;)p Fq(1)2555 4792 y Fu(,)21 b(for)f FE(0)e Ft(<)g Fv(a)h Ft(<)f FE(1)p Fu(,)i Fv(g)e Fj(2)d Fu(C)3273 4761 y Fq(1)p Fl(;)p Fr(a)p Fl(=)p Fq(2)3425 4792 y Fu(,)21 b(and)e(for)520 4914 y Fv(a)g Fk(=)e FE(0)k Fu(then)e(ther)m(e)i(ar)m (e)f(e)n(xamples)g(when)g Fv(g)h Fu(has)f(not)g(a)g(continuous)e (derivative)i(pr)l(o)o(vided)f(that)h FE(dim)9 b Fu(M)22 b Ft(>)c FE(1)p Fu(.)p eop %%Page: 3 3 3 2 bop 488 208 a Fg(DIFFERENTIABILITY)18 b(OF)h(MINIMAL)f(GEODESICS)h (IN)g(METRICS)f(OF)h(LO)n(W)f(REGULARITY)434 b(3)190 461 y FE(Hence)16 b(it)i(is)g(suf)n(\002cient)e(that)i(the)f(metric)f (is)i(H\366lder)e(continuous)f(of)i(an)o(y)f(positi)n(v)o(e)g(order)g (for)g(a)i(minimal)24 583 y(geodesic)j(to)i(ha)n(v)o(e)f(a)h (continuous)d(tangent,)i(and)g(therefore)f(a)i(well-de\002ned)e (arc-length.)f(The)i(assumption)24 705 y(that)17 b(the)g(geodesic)f(is) h(minimal)g(crucial)f(for)g(the)h(theorem)f(and)g(the)h(proof)e(will)j (e)o(xploit)e(this)h(f)o(act)g(repeatedly)-5 b(.)190 827 y(The)28 b(moti)n(v)n(ation)f(for)g(this)i(study)f(comes)h(from)e (the)i(theory)e(of)h(the)g(Schr\366dinger)e(equation.)h(There)24 949 y(an)g(important)g(tool)g(is)i(the)f(Agmon-Lithner)c(metric)k Fu(g)22 b Fk(=)1794 878 y Fp(p)p 1877 878 435 4 v 71 x FE(max)o Fk(\()p Fu(v)p Fk(\()p Fu(x)p Fk(\))p Ft(;)9 b Fu(E)d Fk(\))19 b Fu(d)t(x)2413 919 y Fq(2)2477 949 y FE(where)27 b Fu(v)h FE(is)h(a)f(potential)24 1071 y(function)23 b(and)i(the)h(non-ne)o(gati)n(v)o(e)c Fu(E)32 b 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Fj(\000)g Fu(c)p Fj(j)896 1041 y(\024)989 984 y Fk(\()p FE(1)g Fk(+)g Fv(d)p Fk(\))d(\()o Fu(d)t Fk(\()p Fu(a)p Ft(;)g Fu(b)p Fk(\))j(+)g Fu(d)t Fk(\()p Fu(b)p Ft(;)d Fu(c)p Fk(\)\))1831 946 y Fq(2)1877 984 y Fj(\000)j Fu(d)t Fk(\()p Fu(a)p Ft(;)d Fu(c)p Fk(\))2175 954 y Fq(2)2209 984 y Ft(=)p Fk(\()p FE(1)j Fk(+)g Fv(d)p Fk(\))p 989 1021 V 1355 1097 a FE(2)p Fu(d)t Fk(\()p Fu(a)p Ft(;)d Fu(b)p Fk(\))p Fu(d)t Fk(\()p Fu(b)p Ft(;)g Fu(c)p Fk(\))p Ft(=)p Fk(\()p FE(1)j Fk(+)g Fv(d)p Fk(\))2514 1041 y(=)2596 973 y Fp(\000)2635 1041 y Fv(d)g Fk(+)g Fv(d)2806 1006 y Fq(2)2839 1041 y Ft(=)p FE(2)2923 973 y Fp(\001)2979 984 y Fk(\()p Fu(d)t Fk(\()p Fu(a)p Ft(;)d Fu(b)p Fk(\))j(+)g Fu(d)t Fk(\()p Fu(b)p Ft(;)d Fu(c)p Fk(\)\))3578 946 y Fq(2)p 2979 1021 634 4 v 3073 1097 a Fu(d)t Fk(\()p Fu(a)p Ft(;)g Fu(b)p Fk(\))p Fu(d)t Fk(\()p Fu(b)p Ft(;)g Fu(c)p Fk(\))3622 1041 y Ft(;)520 1249 y FE(since)26 b Fu(d)t Fk(\()p Fu(a)p Ft(;)9 b Fu(c)p Fk(\))21 b(=)g Fu(d)t Fk(\()p Fu(a)p Ft(;)9 b Fu(b)p 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Fn(\345)1953 4785 y Fm(j)r Fi(=)p Fq(0)2060 4699 y FE(2)2102 4665 y Fs(\000)9 b Fm(j)2181 4699 y Fu(g)2223 4711 y Fm(a)2261 4722 y Fd(j)520 4914 y FE(is)21 b(a)f(continuous)f(metric)g(where)h(a)h(gi)n (v)o(en)e(minimal)g(geodesic)g(is)i(not)f(e)n(v)o(en)g(piece)n(wise)g (dif)n(ferentiable.)p eop %%Page: 5 5 5 4 bop 488 208 a Fg(DIFFERENTIABILITY)18 b(OF)h(MINIMAL)f(GEODESICS)h (IN)g(METRICS)f(OF)h(LO)n(W)f(REGULARITY)434 b(5)190 461 y FE(The)28 b(e)o(xample)f(is)j(easily)f(generalized)e(to)i Fh(R)1525 431 y Fm(n)1595 461 y FE(for)f Fu(n)23 b Ft(>)g FE(2)29 b(and)f(by)g(w)o(orking)f(in)i(a)g(local)g(coordinate)24 583 y(system)20 b(to)g(an)o(y)g(\002nite)g(dimensional)f(manifold)f (with)j(dimension)d(greater)i(than)f(one.)190 769 y Fu(Pr)l(oof)h(of)g (Pr)l(oposition)f(2.2.)40 b FE(Let)25 b Fv(g)p Fk(\()n Fu(t)1279 781 y Fq(0)1314 769 y Fk(\))g FE(be)f(an)h(interior)e(point)g (on)h Fv(g)p FE(.)h(W)-7 b(e)25 b(may)f(choose)g(local)g(coordi-)24 891 y(nates)d Fu(x)e Fk(=)f(\()p Fu(x)424 903 y Fq(1)459 891 y Ft(;)9 b(:)g(:)g(:)h(;)f Fu(x)657 903 y Fm(n)692 891 y Fk(\))22 b FE(centred)d(at)j Fv(g)p Fk(\()n Fu(t)1180 903 y Fq(0)1215 891 y Fk(\))g FE(such)e(that)h Fu(g)g FE(is)h(Euclidean)e(at)h Fu(T)2203 908 y Fr(g)p Fi(\()n Fm(t)2267 920 y Fc(0)2297 908 y Fi(\))2325 891 y Fu(M)s FE(.)g(First)h(we)f(study)g(the)g(right)24 1013 y(deri)n(v)n(ati)n(v)o (e)d(of)i Fv(g)g FE(at)f Fu(t)615 1025 y Fq(0)650 1013 y FE(.)h(Let)f Fu(t)24 b Ft(>)18 b FE(0)i(be)g(so)g(small)h(that)f Fv(g)p Fk(\()n Fu(t)1639 1025 y Fq(0)1686 1013 y Fk(+)9 b Fu(t)d Fk(\))20 b FE(is)i(still)f(an)f(interior)f(point)g(on)h Fv(g)p FE(,)h(and)e(so)i(small)24 1135 y(that)f(we)h(do)g(not)f(lea)n (v)o(e)g(the)h(local)f(coordinate)f(patch)h(in)h(the)g(follo)n(wing.)d (Since)j Fu(g)g FE(is)g(H\366lder)f(continuous)e(of)24 1257 y(order)h Fv(a)i FE(we)f(get)g(the)h(estimate)1037 1433 y Fu(d)t(x)1120 1399 y Fq(2)1154 1433 y Ft(=)p Fk(\()p FE(1)12 b Fk(+)c Fu(Ct)1439 1399 y Fr(a)1480 1433 y Fk(\))19 b Ft(<)e Fu(g)h 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y Fs(\000)p Fq(1)2336 3186 y Fu(t)6 b Fk(\))29 b FE(sho)n(ws)f(that)g(the)h(angle)f(at)24 3308 y Fv(g)p Fk(\()n Fu(t)111 3320 y Fq(0)160 3308 y Fk(+)14 b FE(2)281 3278 y Fs(\000)p Fq(1)361 3308 y Fu(t)6 b Fk(\))27 b FE(is)h Fv(p)14 b Fk(+)g Fu(O)p Fk(\()n Fu(t)790 3278 y Fr(a)p Fl(=)p Fq(2)894 3308 y Fk(\))p FE(.)28 b(Hence)e(the)i(angle)e(between)g(the)i(tw)o(o)f(chords)f (through)f Fv(g)p Fk(\()n Fu(t)2758 3320 y Fq(0)2793 3308 y Fk(\))j FE(is)g Fu(O)p Fk(\()n Fu(t)3055 3278 y Fr(a)p Fl(=)p Fq(2)3160 3308 y Fk(\))p FE(.)24 3431 y(Repeating)i(this)j(ar)o(gument)c(with)j(the)f(triangle)g(with)h (corners)f Fv(g)p Fk(\()n Fu(t)2032 3443 y Fq(0)2067 3431 y Fk(\))p FE(,)h Fv(g)p Fk(\()n Fu(t)2239 3443 y Fq(0)2290 3431 y Fk(+)15 b FE(2)2412 3400 y Fs(\000)p Fm(k)2490 3431 y Fu(t)6 b Fk(\))32 b FE(and)f Fv(g)p Fk(\()n Fu(t)2822 3443 y Fq(0)2873 3431 y Fk(+)16 b FE(2)2996 3400 y Fs(\000)p Fm(k)q Fs(\000)p Fq(1)3152 3431 y Fu(t)6 b Fk(\))24 3553 y FE(sho)n(ws)19 b(that)h(the)g(angle)f(between)g(the)h (tw)o(o)g(chords)f(through)e Fv(g)p Fk(\()n Fu(t)1886 3565 y Fq(0)1922 3553 y Fk(\))j FE(is)h Fu(O)p Fk(\()p FE(2)2184 3522 y Fs(\000)p Fm(k)q Fr(a)p Fl(=)p Fq(2)2362 3553 y Fu(t)2391 3522 y Fr(a)p Fl(=)p Fq(2)2495 3553 y Fk(\))p FE(.)f(Holding)d Fu(t)26 b FE(\002x)o(ed)19 b(and)24 3675 y(letting)e Fu(k)h Fj(!)f Fv(\245)h FE(we)h(see)f(that)h Fk(\()p Fv(g)p Fk(\()n Fu(t)989 3687 y Fq(0)1034 3675 y Fk(+)10 b FE(2)1151 3644 y Fs(\000)p Fm(k)1227 3675 y Fu(t)c Fk(\))k Fj(\000)g Fv(g)p Fk(\()n Fu(t)1460 3687 y Fq(0)1494 3675 y Fk(\)\))p Ft(=)p Fk(\()p FE(2)1674 3644 y Fs(\000)p Fm(k)1752 3675 y Fu(t)c Fk(\))18 b FE(has)h(a)f(limit) g(v)o(ector)f(of)h(length)f(one.)g(This)i(limit)24 3797 y(is)j(independent)d(of)i(the)g(choice)f(of)g Fu(t)27 b FE(which)21 b(is)h(easily)g(seen)f(by)g(mixing)f(tw)o(o)i(dif)n (ferent)d(sequences.)i(Hence)24 3919 y Fv(g)58 3889 y Fs(0)79 3919 y Fk(\()n Fu(t)132 3931 y Fq(0)179 3919 y Fk(+)13 b FE(0)p Fk(\))23 b 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y Fk(\()n Fu(t)688 4297 y Fq(0)735 4285 y Fk(+)10 b Fu(t)c Fk(\))20 b FE(e)o(xists)g(for)g(all)f Fu(t)6 b FE(.)190 4407 y(Ne)o(xt)28 b(we)h(study)e(the)i(re)o(gularity) d(of)i Fv(g)1326 4377 y Fs(0)1348 4407 y FE(.)g(T)-7 b(o)29 b(estimate)f(ho)n(w)g(the)h(length)e(of)h Fv(g)2486 4377 y Fs(0)2536 4407 y FE(v)n(aries)g(we)h(get)g(in)f(the)24 4529 y(coordinates)18 b(centred)h(at)i Fv(g)p Fk(\()n Fu(t)860 4541 y Fq(0)895 4529 y Fk(\))g FE(and)f(small)g Fu(s)672 4674 y Fj(j)p Fu(s)p Fj(j)p 586 4712 252 4 v 586 4788 a FE(1)12 b Fk(+)c Fu(Ct)797 4764 y Fr(a)866 4731 y Fk(=)959 4674 y Fu(d)t Fk(\()p Fv(g)p Fk(\()n Fu(t)1124 4686 y Fq(0)1171 4674 y Fk(+)i Fu(t)16 b Fk(+)c Fu(s)p Fk(\))p Ft(;)d Fv(g)p Fk(\()n Fu(t)1545 4686 y Fq(0)1591 4674 y Fk(+)h Fu(t)c Fk(\)\))p 959 4712 800 4 v 1233 4788 a FE(1)12 b Fk(+)c Fu(Ct)1444 4764 y Fr(a)1787 4731 y Fj(\024)18 b(j)p Fv(g)p Fk(\()n Fu(t)1980 4743 y Fq(0)2026 4731 y Fk(+)10 b Fu(t)17 b Fk(+)12 b Fu(s)p Fk(\))g Fj(\000)g Fv(g)p Fk(\()n Fu(t)2458 4743 y Fq(0)2503 4731 y Fk(+)e Fu(t)c Fk(\))p Fj(j)760 4906 y(\024)18 b Fk(\()p FE(1)12 b Fk(+)c Fu(Ct)1086 4872 y Fr(a)1127 4906 y Fk(\))p Fu(d)t Fk(\()p Fv(g)p Fk(\()n Fu(t)1324 4918 y Fq(0)1371 4906 y Fk(+)i Fu(t)16 b Fk(+)c Fu(s)p Fk(\))p Ft(;)d Fv(g)p Fk(\()n Fu(t)1745 4918 y Fq(0)1792 4906 y Fk(+)h Fu(t)c Fk(\)\))17 b(=)h(\()p FE(1)12 b Fk(+)c Fu(Ct)2303 4872 y Fr(a)2344 4906 y Fk(\))h Fj(j)q Fu(s)p Fj(j)p eop %%Page: 6 6 6 5 bop 520 208 a Fg(6)1290 b(PELLE)13 b(PETTERSSON)520 461 y FE(that)20 b(is)1456 561 y(1)p 1351 598 252 4 v 1351 674 a(1)12 b Fk(+)c Fu(Ct)1562 650 y Fr(a)1631 617 y Fj(\024)1714 497 y Fp(\014)1714 547 y(\014)1714 597 y(\014)1714 646 y(\014)1752 561 y Fv(g)p Fk(\()n Fu(t)1839 573 y Fq(0)1886 561 y Fk(+)i Fu(t)16 b Fk(+)c Fu(s)p Fk(\))g Fj(\000)g Fv(g)p Fk(\()n Fu(t)2317 573 y Fq(0)2362 561 y Fk(+)e Fu(t)c Fk(\))p 1752 598 745 4 v 2108 674 a Fu(s)2506 497 y Fp(\014)2506 547 y(\014)2506 597 y(\014)2506 646 y(\014)2553 617 y Fj(\024)17 b FE(1)12 b Fk(+)c Fu(Ct)2846 583 y Fr(a)520 809 y FE(and)19 b(hence)1503 958 y(1)p Ft(=)p Fk(\()p FE(1)12 b Fk(+)c Fu(Ct)1830 923 y Fr(a)1870 958 y Fk(\))19 b Fj(\024)2004 887 y Fp(\014)2004 937 y(\014)2032 958 y Fv(g)2066 923 y Fs(0)2087 958 y Fk(\()n Fu(t)2140 970 y Fq(0)2186 958 y Fk(+)10 b Fu(t)c Fk(\))2322 887 y Fp(\014)2322 937 y(\014)2367 958 y Fj(\024)18 b FE(1)12 b Fk(+)c Fu(Ct)2661 923 y Fr(a)2702 958 y Ft(:)520 1128 y FE(Since)20 b Fj(j)p Fv(g)782 1097 y Fs(0)803 1128 y Fk(\()n Fu(t)856 1140 y Fq(0)891 1128 y Fk(\))p Fj(j)f Fk(=)f FE(1)i(it)h(follo)n(ws)f(that)1307 1343 y Fj(\000)l Fu(Ct)1452 1308 y Fr(a)1513 1343 y Fj(\024)e(\000)1731 1286 y Fu(Ct)1815 1256 y Fr(a)p 1671 1324 252 4 v 1671 1400 a FE(1)12 b Fk(+)c Fu(Ct)1882 1376 y Fr(a)1951 1343 y Fj(\024)2034 1272 y Fp(\014)2034 1322 y(\014)2061 1343 y Fv(g)2095 1308 y Fs(0)2116 1343 y Fk(\()n Fu(t)2169 1355 y Fq(0)2216 1343 y Fk(+)i Fu(t)c Fk(\))2352 1272 y Fp(\014)2352 1322 y(\014)2390 1343 y Fj(\000)2467 1272 y Fp(\014)2467 1322 y(\014)2494 1343 y Fv(g)2528 1308 y Fs(0)2549 1343 y Fk(\()n Fu(t)2602 1355 y Fq(0)2637 1343 y Fk(\))2669 1272 y Fp(\014)2669 1322 y(\014)2715 1343 y Fj(\024)14 b Fu(Ct)2878 1308 y Fr(a)520 1540 y FE(that)20 b(is)1643 1618 y Fp(\014)1643 1668 y(\014)1671 1618 y(\014)1671 1668 y(\014)1698 1689 y Fv(g)1732 1655 y Fs(0)1753 1689 y Fk(\()n Fu(t)1806 1701 y Fq(0)1853 1689 y Fk(+)10 b Fu(t)c Fk(\))1989 1618 y Fp(\014)1989 1668 y(\014)2027 1689 y Fj(\000)2104 1618 y Fp(\014)2104 1668 y(\014)2131 1689 y Fv(g)2165 1655 y Fs(0)2186 1689 y Fk(\()n Fu(t)2239 1701 y Fq(0)2274 1689 y Fk(\))2306 1618 y Fp(\014)2306 1668 y(\014)2334 1618 y(\014)2334 1668 y(\014)2380 1689 y Fj(\024)14 b Fu(Ct)2543 1655 y Fr(a)520 1859 y FE(and)k(hence)g(the)i(length)e(of)h Fv(g)1344 1829 y Fs(0)1384 1859 y FE(is)h(H\366lder)f(continuous)e(of)h (order)g Fv(a)p FE(.)i(But)g(the)f(angle)f(is)i(H\366lder)e(continuous) g(of)520 1981 y(order)k Fv(a)p Ft(=)p FE(2.)48 b(Hence,)23 b(by)h(letting)d Fu(t)26 b Fj(!)21 b FE(0,)i(we)h(conclude)e(that)i Fv(g)2390 1951 y Fs(0)2436 1981 y FE(is)g(H\366lder)f(continuous)f(of)i (order)e Fv(a)p Ft(=)p FE(2)i(for)520 2103 y(the)c(ar)o(gument)e(is)j (uniform)d(in)g Fu(t)1445 2115 y Fq(0)1480 2103 y FE(.)2143 b Ff(\003)686 2306 y FE(When)28 b Fv(a)c Fk(=)f FE(1,)28 b(that)h(is,)g(when)f(the)h(metric)f(is)i(Lipschitz)e(continuous,)f (then)h(the)h(metric)f(is)i(re)o(gular)520 2428 y(enough)18 b(so)i(that)h(re)o(gularization)c(will)k(yield)f(a)h(stronger)d (result.)688 2550 y(P)9 b FF(R)g(O)g(P)g(O)g(S)g(I)g(T)g(I)g(O)g(N)19 b FE(2.4)r(.)43 b Fu(Let)21 b(g)f(be)g(a)g(Lipsc)o(hitz)h(continuous,)d (positive)i(Riemannian)e(metric)j(on)f(M)s(.)h(If)520 2672 y Fk([)p FE(0)p Ft(;)9 b FE(1)p Fk(])18 b Fj(3)f Fu(t)25 b Fj(7!)19 b Fv(g)p Fk(\()n Fu(t)6 b Fk(\))23 b Fu(is)f(a)f(minimal)h(g)o(eodesic)e(with)i(the)g(par)o(ameter)e(pr)l (oportional)g(to)i(the)f(ar)m(c-length,)f(then)h Fv(g)520 2794 y Fu(is)g(in)16 b(C)734 2764 y Fq(1)790 2794 y Fu(with)21 b(Lipsc)o(hitz)f(continuous)e(\002r)o(st)j(derivative)o(.)686 2916 y FE(This)32 b(generalizes)f(a)i(corresponding)c(result)j(for)f (geodesics)h(on)g(a)g(2-dimensional)e(surf)o(ace)i(in)g Fh(R)3668 2886 y Fq(3)520 3038 y FE(in)20 b([4)o(].)686 3241 y Fu(Pr)l(oof)o(.)40 b FE(In)19 b(the)g(metric)33 b(\210)-42 b Fu(g)17 b Fk(=)h(\()p FE(1)11 b Fk(+)g Fv(r)p Fk(\))p Fu(g)p FE(,)17 b(where)i Fv(r)g FE(is)h(a)g(non-ne)o(gati)n(v)o (e,)15 b(smooth)j(function)g(v)n(anishing)g(on)520 3363 y Fv(g)p Fk(\([)p FE(0)p Ft(;)9 b FE(1)p Fk(]\))24 b FE(and)f(positi)n(v)o(e)h(on)f Fu(M)17 b Fj(n)c Fv(g)p Fk(\([)p FE(0)p Ft(;)c FE(1)p Fk(]\))p FE(,)23 b Fv(g)i FE(is)g(the)f(unique,)e(minimal)i(geodesic)f(between)g Fv(g)p Fk(\()p FE(0)p Fk(\))h FE(and)g Fv(g)p Fk(\()p FE(1)p Fk(\))p FE(.)520 3485 y(Such)19 b(a)h(function)e Fv(r)i FE(can)g(be)f(constructed,)f(for)h(e)o(xample,)f(by)i(starting)f (with)h(a)g(partition)f(of)g(unity)3402 3491 y Fv(\345)3471 3485 y(q)3523 3497 y Fm(j)3564 3485 y Fk(=)e FE(1,)520 3607 y Fv(q)572 3619 y Fm(j)617 3607 y Fj(2)i Fu(C)748 3577 y Fr(\245)744 3631 y Fq(0)796 3607 y Fk(\()p Fu(M)f Fj(n)13 b Fv(g)p Fk(\([)p FE(0)p Ft(;)c FE(1)p Fk(]\)\))28 b FE(where)f(the)g(sum)g(is)h(locally)f(\002nite)h(and)e(then)h (setting)g Fv(r)22 b Fk(=)3074 3613 y Fv(\345)3143 3607 y(e)3188 3619 y Fm(j)3211 3607 y Fv(q)3263 3619 y Fm(j)3314 3607 y FE(with)27 b Fv(e)3534 3619 y Fm(j)3580 3607 y Ft(>)22 b FE(0)520 3729 y(v)n(anishing)j(f)o(ast)i(enough.)e(The)h (metric)40 b(\210)-41 b Fu(g)26 b FE(is)i(still)g(Lipschitz)e (continuous.)f(If)h Fu(L)i FE(is)g(the)f(length)e(of)i Fv(g)g FE(in)g(the)520 3851 y(metric)34 b(\210)-41 b Fu(g)21 b FE(and)g Fv(W)h FE(is)h(an)e(open,)f(bounded)f(neighbourhood) e(of)k Fv(g)p FE(,)h(then)f(there)g(e)o(xists)h(a)g(constant)f Fv(d)e Ft(>)g FE(0)i(such)520 3973 y(that)e(all)i(curv)o(es)d(from)h Fv(g)p Fk(\()p FE(0)p Fk(\))h FE(to)g Fv(g)p Fk(\()p FE(1)p Fk(\))g FE(that)g(lea)n(v)o(e)f Fv(W)h FE(ha)n(v)o(e)f(a)h (length)f(at)h(least)g Fk(\()p FE(1)11 b Fk(+)g Fv(d)p Fk(\))p Fu(L)p FE(.)40 b(Since)p 3268 3907 64 4 v 20 w Fv(W)20 b FE(is)g(compact)520 4095 y(and)32 b(\210)-41 b Fu(g)21 b FE(is)g(Lipschitz)e(continuous)g(there)g(is)i(a)g(constant) 16 b Fu(C)2181 4107 y Fq(1)2236 4095 y FE(such)k(that)1482 4287 y Fj(j)14 b FE(\210)-42 b Fu(g)p Fk(\()p Fu(x)12 b Fk(+)g Fu(z)p Fk(\))g Fj(\000)23 b FE(\210)-39 b Fu(g)m Fk(\()p Fu(x)p Fk(\))p Fj(j)19 b(\024)14 b Fu(C)2172 4299 y Fq(1)2217 4287 y Fj(j)p Fu(z)p Fj(j)9 b Ft(;)175 b Fj(j)p Fu(z)p Fj(j)19 b Ft(<)f Fv(d)p Ft(:)520 4478 y FE(If)k Fv(c)d Fj(2)d Fu(C)791 4448 y Fr(\245)787 4502 y Fq(0)839 4478 y Fk(\()p Fu(M)s Fk(\))p FE(,)24 b Fv(c)19 b Fj(\025)h FE(0)i(with)1405 4416 y Fo(R)1465 4478 y Fv(c)p Fk(\()p Fu(y)p Fk(\))9 b Fu(d)t(y)20 b Fk(=)f FE(1,)j(then)g(let)h Fv(c)2211 4490 y Fr(e)2242 4478 y Fk(\()p Fu(y)p Fk(\))d(=)f Fv(e)2483 4448 y Fs(\000)p Fm(n)2566 4478 y Fv(c)p Fk(\()p Fu(y)p Ft(=)p Fv(e)p Fk(\))k FE(be)f(a)h(standard)f(molli\002er)g(and)520 4600 y(let)31 b(\210)-41 b Fu(g)663 4570 y Fr(e)710 4600 y Fk(=)29 b FE(\210)-41 b Fu(g)8 b Fj(\003)h Fv(c)938 4612 y Fr(e)987 4600 y FE(be)18 b(a)g(standard)f(re)o(gularization)e (of)31 b(\210)-41 b Fu(g)17 b FE(such)h(that)31 b(\210)-41 b Fu(g)2420 4570 y Fr(e)2467 4600 y Fj(2)12 b Fu(C)2591 4570 y Fr(\245)2658 4600 y FE(and)17 b(has)h(uniformly)d(bounded)g (\002rst)520 4722 y(deri)n(v)n(ati)n(v)o(es)j(in)j Fv(W)f FE(and)g(satis\002es)1644 4914 y(\210)-41 b Fu(g)p Ft(=)p Fk(\()p FE(1)12 b Fk(+)c Fu(C)r Fv(e)p Fk(\))17 b Fj(\024)31 b FE(\210)-41 b Fu(g)2141 4879 y Fr(e)2190 4914 y Fj(\024)31 b FE(\210)-41 b Fu(g)o Fk(\()p FE(1)12 b Fk(+)c Fu(C)r Fv(e)p Fk(\))p eop %%Page: 7 7 7 6 bop 488 208 a Fg(DIFFERENTIABILITY)18 b(OF)h(MINIMAL)f(GEODESICS)h (IN)g(METRICS)f(OF)h(LO)n(W)f(REGULARITY)434 b(7)24 461 y FE(for)19 b(some)d Fu(C)21 b Ft(>)d FE(0.)i(In)g(f)o(act,)g(we)g(ha)n (v)o(e)g(to)g(estimate)153 647 y Fv(\266)194 659 y Fm(x)228 670 y Fd(j)266 647 y FE(\210)-42 b Fu(g)294 613 y Fr(e)344 647 y Fk(=)18 b Fv(\266)468 659 y Fm(x)502 670 y Fd(j)551 591 y FE(1)p 536 628 72 4 v 536 704 a Fv(e)572 680 y Fm(n)626 544 y Fo(Z)725 647 y FE(\210)-41 b Fu(g)o Fk(\()p Fu(y)p Fk(\))p Fv(c)900 555 y Fp(\020)960 591 y Fu(x)12 b Fj(\000)g Fu(y)p 960 628 162 4 v 1022 704 a Fv(e)1131 555 y Fp(\021)1190 647 y Fu(d)t(y)18 b Fk(=)1438 591 y FE(1)p 1384 628 150 4 v 1384 704 a Fv(e)1420 680 y Fm(n)p Fi(+)p Fq(1)1553 544 y Fo(Z)1652 647 y FE(\210)-41 b Fu(g)p Fk(\()p Fu(y)p Fk(\))p Fv(c)1828 613 y Fs(0)1837 668 y Fm(j)1859 555 y Fp(\020)1919 591 y Fu(x)12 b Fj(\000)g Fu(y)p 1919 628 162 4 v 1981 704 a Fv(e)2090 555 y Fp(\021)2149 647 y Fu(d)t(y)18 b Fk(=)2343 591 y FE(1)p 2343 628 42 4 v 2346 704 a Fv(e)2404 544 y Fo(Z)2503 647 y FE(\210)-41 b Fu(g)o Fk(\()p Fu(x)12 b Fj(\000)g Fv(e)p Fu(y)p Fk(\))p Fv(c)2840 613 y Fs(0)2849 668 y Fm(j)2871 647 y Fk(\()p Fu(y)p Fk(\))d Fu(d)t(y)-5 b Ft(:)24 846 y FE(Observing)18 b(that)545 813 y Fq(1)p 545 827 31 4 v 547 874 a Fr(e)595 784 y Fo(R)669 846 y FE(\210)-41 b Fu(g)o Fk(\()p Fu(x)p Fk(\))p Fv(c)844 816 y Fs(0)853 869 y Fm(j)876 846 y Fk(\()p Fu(y)p Fk(\))9 b Fu(d)t(y)19 b Fk(=)1181 813 y Fq(1)p 1181 827 V 1183 874 a Fr(e)1235 846 y FE(\210)-41 b Fu(g)o Fk(\()p Fu(x)p Fk(\))1373 784 y Fo(R)1434 846 y Fv(c)1480 816 y Fs(0)1489 869 y Fm(j)1512 846 y Fk(\()p Fu(y)p Fk(\))9 b Fu(d)t(y)18 b Fk(=)g FE(0)j(we)f(get)686 994 y Fp(\014)686 1044 y(\014)713 1064 y Fv(\266)754 1076 y Fm(x)788 1087 y Fd(j)825 1064 y FE(\210)-41 b Fu(g)854 1030 y Fr(e)885 1064 y Fk(\()p Fu(x)p Fk(\))986 994 y Fp(\014)986 1044 y(\014)1033 1064 y Fk(=)1116 944 y Fp(\014)1116 994 y(\014)1116 1044 y(\014)1116 1094 y(\014)1153 1008 y FE(1)p 1153 1045 42 4 v 1156 1121 a Fv(e)1214 961 y Fo(Z)1313 1064 y FE(\210)g Fu(g)o Fk(\()p Fu(x)12 b Fj(\000)g Fv(e)p Fu(y)p Fk(\))p Fv(c)1650 1030 y Fs(0)1659 1085 y Fm(j)1681 1064 y Fk(\()p Fu(y)p Fk(\))d Fu(d)t(y)j Fj(\000)1972 1008 y FE(1)p 1972 1045 V 1975 1121 a Fv(e)2033 961 y Fo(Z)2132 1064 y FE(\210)-42 b Fu(g)p Fk(\()p Fu(x)p Fk(\))p Fv(c)2307 1030 y Fs(0)2316 1085 y Fm(j)2339 1064 y Fk(\()p Fu(y)p Fk(\))9 b Fu(d)t(y)2532 944 y Fp(\014)2532 994 y(\014)2532 1044 y(\014)2532 1094 y(\014)861 1294 y Fj(\024)954 1238 y FE(1)p 954 1275 V 957 1351 a Fv(e)1014 1190 y Fo(Z)1100 1294 y Fj(j)k FE(\210)-41 b Fu(g)p Fk(\()p Fu(x)12 b Fj(\000)g Fv(e)p Fu(y)p Fk(\))g Fj(\000)23 b FE(\210)-39 b Fu(g)m Fk(\()p Fu(x)p Fk(\))p Fj(j)1690 1223 y Fp(\014)1690 1273 y(\014)1718 1294 y Fv(c)1764 1260 y Fs(0)1773 1314 y Fm(j)1795 1294 y Fk(\()p Fu(y)p Fk(\))1896 1223 y Fp(\014)1896 1273 y(\014)1943 1294 y Fu(d)t(y)861 1507 y Fj(\024)954 1450 y FE(1)p 954 1488 V 957 1564 a Fv(e)1001 1507 y Fu(C)1054 1519 y Fq(1)1089 1507 y Fv(e)1134 1403 y Fo(Z)1221 1507 y Fj(j)p Fu(y)p Fj(j)1313 1436 y Fp(\014)1313 1486 y(\014)1340 1507 y Fv(c)1386 1472 y Fs(0)1395 1527 y Fm(j)1418 1507 y Fk(\()p Fu(y)p Fk(\))1519 1436 y Fp(\014)1519 1486 y(\014)1565 1507 y Fu(d)t(y)19 b Fk(=)13 b Fu(C)1798 1519 y Fq(1)1830 1507 y Fu(C)1883 1519 y Fq(2)24 1705 y FE(for)20 b(all)h Fu(x)e Fj(2)g Fv(W)p FE(.)64 b(Furthermore,)18 b(since)34 b(\210)-41 b Fu(g)21 b FE(is)g(positi)n(v)o(e)f(and)p 1736 1638 64 4 v 21 w Fv(W)h FE(is)h(compact)d(there)h(e)o(xists)i(a)f (positi)n(v)o(e)f(constant)24 1827 y Fu(c)g FE(such)g(that)34 b(\210)-42 b Fu(g)18 b Fj(\025)g Fu(c)h Ft(>)f FE(0)i(in)p 828 1760 V 20 w Fv(W)p FE(.)h(Then)e(we)h(ha)n(v)o(e)305 2026 y Fj(j)13 b FE(\210)-41 b Fu(g)370 1992 y Fr(e)400 2026 y Fk(\()p Fu(x)p Fk(\))12 b Fj(\000)25 b FE(\210)-41 b Fu(g)o Fk(\()p Fu(x)p Fk(\))p Fj(j)19 b Fk(=)857 1906 y Fp(\014)857 1956 y(\014)857 2006 y(\014)857 2055 y(\014)885 1923 y Fo(Z)984 2026 y FE(\210)-42 b Fu(g)p Fk(\()p Fu(x)12 b Fj(\000)g Fv(e)p Fu(y)p Fk(\))p Fv(c)p Fk(\()p Fu(y)p Fk(\))d Fu(d)t(y)j Fj(\000)1603 1923 y Fo(Z)1700 2026 y FE(\210)-41 b Fu(g)o Fk(\()p Fu(x)p Fk(\))p Fv(c)p Fk(\()p Fu(y)p Fk(\))9 b Fu(d)t(y)2068 1906 y Fp(\014)2068 1956 y(\014)2068 2006 y(\014)2068 2055 y(\014)480 2256 y Fj(\024)563 2152 y Fo(Z)649 2256 y Fj(j)k FE(\210)-41 b Fu(g)o Fk(\()p Fu(x)12 b Fj(\000)g Fv(e)p Fu(y)p Fk(\))g Fj(\000)23 b FE(\210)-39 b Fu(g)n Fk(\()p Fu(y)p Fk(\))p Fj(j)9 b(j)p Fv(c)p Fk(\()p Fu(y)p Fk(\))p Fj(j)19 b Fu(d)t(y)f Fj(\024)c Fu(C)1683 2268 y Fq(1)1727 2152 y Fo(Z)1813 2256 y Fv(e)9 b Fj(j)p Fu(y)p Fj(j)h(j)p Fv(c)p Fk(\()p Fu(y)p Fk(\))p Fj(j)18 b Fu(d)t(y)g Fk(=)c Fu(C)2395 2268 y Fq(1)2426 2256 y Fu(C)2479 2268 y Fq(2)2514 2256 y Fv(e)19 b Fj(\024)2658 2200 y Fu(C)2711 2212 y Fq(1)2742 2200 y Fu(C)2795 2212 y Fq(2)2830 2200 y Fv(e)p 2662 2237 205 4 v 2746 2313 a Fu(c)2889 2256 y FE(\210)-41 b Fu(g)p Ft(:)24 2443 y FE(W)m(ith)16 b Fu(C)r Ft(=)p FE(2)i Fk(=)c Fu(C)494 2455 y Fq(1)525 2443 y Fu(C)578 2455 y Fq(2)613 2443 y Ft(=)p Fu(c)20 b FE(we)g(get)1061 2619 y Fk(\()p FE(1)12 b Fj(\000)c Fu(C)r Fv(e)p Ft(=)p FE(2)p Fk(\))13 b FE(\210)-41 b Fu(g)16 b Fj(\024)31 b FE(\210)-41 b Fu(g)1612 2584 y Fr(e)1661 2619 y Fj(\024)18 b Fk(\()p FE(1)12 b Fk(+)c Fu(C)r Fv(e)p Ft(=)p FE(2)p Fk(\))13 b FE(\210)-41 b Fu(g)n 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Fr(e)1531 4393 y Fk(\()p Fu(s)p Fk(\))1627 4322 y Fp(\014)1627 4372 y(\014)1674 4393 y Fj(\024)i Fu(C)1806 4405 y Fq(3)1850 4393 y Fj(j)n Fu(t)j Fj(\000)12 b Fu(s)p Fj(j)24 4548 y FE(for)23 b(all)h Fu(s)p FE(,)e Fu(t)30 b FE(and)23 b Fv(e)p FE(.)h(Hence)f Fv(g)875 4517 y Fs(0)875 4568 y Fr(e)930 4548 y FE(is)i(equicontinuous.)20 b(Then)j(the)g(Arzela-Ascoli)g(theorem)f(sho)n(ws)i(that)g(there)24 4670 y(is)k(a)f(subsequence)e(such)i(that)g Fv(g)984 4682 y Fr(e)1043 4670 y FE(con)m(v)o(er)o(ges)d(to)j(a)h(function)d Fk([)p FE(0)p Ft(;)9 b FE(1)p Fk(])21 b Fj(3)g Fu(t)28 b Fj(7!)22 b Fv(h)p Fk(\()n Fu(t)6 b Fk(\))28 b FE(that)f(belongs)f(to) d Fu(C)3029 4639 y Fq(1)3064 4670 y Fk(\()p Fv(W)p Fk(\))p FE(.)24 4792 y(Going)e(to)i(the)g(limit)g(in)f(the)h(inequality)e(abo)o (v)o(e)g(sho)n(ws)i(that)f Fv(h)h FE(has)g(Lipschitz)f(continuous)f (\002rst)i(deri)n(v)n(ati)n(v)o(e)24 4914 y(and)e(since)35 b(\210)-41 b Fu(g)o Fk(\()p Fv(g)466 4884 y Fs(0)466 4934 y Fr(e)498 4914 y Fk(\()n Fu(t)6 b Fk(\))p Ft(;)j Fv(g)655 4884 y Fs(0)655 4934 y Fr(e)686 4914 y Fk(\()n Fu(t)d Fk(\)\))20 b(=)f FE(1)j(for)f(all)f Fu(t)28 b FE(and)21 b Fv(e)i FE(it)g(follo)n(ws)e(that)35 b(\210)-41 b Fu(g)p Fk(\()p Fv(h)2060 4884 y Fs(0)2081 4914 y Fk(\()n Fu(t)6 b Fk(\))p Ft(;)j Fv(h)2254 4884 y Fs(0)2275 4914 y Fk(\()n Fu(t)d Fk(\)\))20 b(=)f FE(1.)44 b(Hence)21 b Fv(h)p Fk(\()n Fu(t)6 b Fk(\))20 b(=)e Fv(g)p Fk(\()n Fu(t)6 b Fk(\))p eop %%Page: 8 8 8 7 bop 520 208 a Fg(8)1290 b(PELLE)13 b(PETTERSSON)520 461 y FE(for)18 b Fu(t)24 b Fk(=)18 b FE(0)p Ft(;)9 b FE(1,)21 b(and)f(the)g(length)g(with)h(respect)f(to)h(the)g(metric)34 b(\210)-42 b Fu(g)21 b FE(is)h(at)f(most)g Fu(L)p FE(.)g(Since)g(by)f (the)h(construction)e(of)533 583 y(\210)-41 b Fu(g)o FE(,)21 b Fv(g)g FE(is)h(the)f(unique)e(minimal)h(geodesic)f(between)h Fv(g)p Fk(\()p FE(0)p Fk(\))h FE(and)f Fv(g)p Fk(\()p FE(1)p Fk(\))p FE(,)h(and)f(it)i(follo)n(ws)e(that)g Fv(g)p Fk(\()n Fu(t)6 b Fk(\))19 b(=)f Fv(h)p Fk(\()n Fu(t)6 b Fk(\))22 b 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