SYMMETRIZED CURVE-STRAIGHTENING

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1 SYMMETRIZED CURVE-STRIGTENING NDERS LINNÉR STRCT. The rdiionl curve-righening low i ed on one o he ndrd Soolev inner produc nd i i known o rek cerin ymmerie o relecion. The purpoe o hi pper i o how h here re lernive Riemnnin rucure on he pce o curve h yield low h preerve ymmerie. Thi eure come price. In one ymmerizing meric he grdien vecor ield re coniderly more demnding o compue. In noher ymmerizing meric moohne i lo. Thi inveigion will lo explin he phenomen o pinning oerved in everl exmple in he rdiionl low. Three cle o lernive Riemnnin rucure re exmined. The ir cl include he rdiionl meric pecil ce nd i hown o never preerve oh roion ymmerie nd ymmerie o relecion. The econd cl coni o ingle meric correponding o one o he ndrd Soolev meric, nd i hown o preerve oh ype o ymmerie. The hird cl lo include he rdiionl meric u i i hown h here i unique dieren meric in hi cl, which preerve oh ype o ymmerie. Thi priculr meric generlly yield mooh vecor ield, which when evlued mooh uncion do no give mooh elemen o he correponding ngen pce. The hird cl i neverhele preerred ince i h he diincion h i repec he projecion induced y he derivive operor ono he ngen undle o he pce o derivive. The pper conclude wih numer o grphicl illurion h how preerved ymmery nd removl o pinning.. INTRODUCTION. ckground. Conider Riemnnin mniold X wih meric,, nd uncionl F : X. The direcionl derivive DF( x) : Tx X i reled o he grdien Fx ( ) TX x y Fx ( ), v = DFxv ( ). The curve-righening low wih ν i he x negive grdien low ocied wih he modiied ol qured curvure uncionl γ + ν. γ k d lhough here i cerin moun o reedom in he choice o X nd i Riemnnin rucure, pper deling wih he curve-righening low proper hve o r conienly ued he me pce X nd meric. Speciiclly, X i he lrge poile Soolev pce on which he modiied

2 NDERS LINNÉR ol qured curvure uncionl i deined wih he imple ndrd meric. Thi choice o cilie he compuion o he grdien nd i projecion ono ngen pce in he ce o conrin. I lo impliie everl o he nlyicl rgumen required when veriying he Pli-Smle condiion.. Purpoe. The purpoe o hi pper i o nwer queion concerning he ilure o he rdiionl curve-righening low o preerve cerin ymmerie o relecion. There re mny exmple h illure hi ehvior; ee []. Roer L. ryn nd Joel Lnger hve on epre occion ked he uhor why he ymmery i roken. The hor, u no priculrly iying nwer, i h he meric vw, v() w() vwd ( ) ( ) = + i no ymmeric on he prmeer inervl [, ]. In he ce o cloed curve mo o he eec o hi ymmeric choice re mked y he periodiciy.. Prolem. Recll h oluion o he equion F( x) = re clled criicl poin. The choice o meric ec he orm o he equion F( x) =. criicl poin iie DF ( x ) v = or ll v. I ollow h he choice o meric i no imporn when he min concern i o ind ll oluion o Fx ( ) =. The lck o ymmery in he meric, ecome n iue when he concern i he ymmery o x τ in he low dxτ F( xτ ) dτ =. reled queion, poed y Dvid. Singer, concern n exmple given in []. The exmple how he curve-righening low o n iniil curve, which i lmo doule covered Euler igure eigh. The low h limi he ingle covered Euler igure eigh. The purpoe o he exmple i o how complee negive grdien rjecory, which r cloe o he unle doule covered Euler igure eigh nd end he le ingle covered Euler igure eigh. The iniil curve h n xi o ymmery nd uequen curve long he rjecory lo hve n xi o ymmery. The peculir ehvior i h he xi o ymmery roe during he low nd Dvid queion i why hi i he ce. The reul in [] concerning preerved ymmery do no pply o hi ce ince he roion numer in he exmple i oppoed o. The roo cue o he ehvior i he exr degree o reedom due o he poiiliy o pinning cloed curve. In Secion.4 he echnicl mniold heoreic reon ehind pinning i expoed..4 Orgnizion. In Secion ligh exenion o he concep o odd nd even uncion i inroduced. Thi led o he noion o prmeric ymmerie. In he conex o curve in he plne hi exenion eicienly ideniie non-cloed curve wih n xi o ymmery or 8 degree roionl ymmery. In he ce o cloed curve he prmeric ymmerie re only preen in ome o he prmeerizion o he choice o iniil poin i in hi ce imporn. The pce

3 SYMMETRIZED CURVE-STRIGTENING o prmericlly ymmeric curve re nlyzed ue in he pce o ll curve. In [] exernlly ymmeric cloed curve o roion numer re hown o e cloed umniold uing echnique y Pli. The exernlly ymmeric curve re no o e conued wih he prmericlly ymmeric curve conidered here. In Secion he generl ormul derived in [4] re reclled nd pecilized o he conex o he curve-righening low. Unconrined well conrined curve re conidered. The min reul i he proo o he exience o curverighening low h preerve ymmerie. To keep he preen pper el-conined n ppendix i included wih direc proo o he generl ormul derived in [4].. PRMETRIC SYMMETRIES. Generlized even nd odd. The concep o even nd odd re ued here in lighly more generl eing. Given n inervl [, ], le m = ( + )/ e he midpoin. uncion : [, ] i generlized even wih repec o he midpoin i ( x) = ( + x) or ll x [, ]. The uncion i generlized odd wih repec o he midpoin i ( x) + ( + x) = ( m). Noe h only he conn uncion hve oh ymmerie.. Prmeric ymmery ou poin. Suppoe regulr (i.e., dierenile nd nonzero peed) curve γ : [,] o lengh L in he complex plne i reled o coninuou uncion : [,] y i ( ) γ ( ) = Le. The curve γ i id o hve prmeric roionl ymmery wih repec o he poin γ ( /) i he ngen ngle i generlized even wih repec o /. The curve γ i id o hve prmeric relecion ymmery wih repec o he norml line hrough γ ( /) i i generlized odd wih repec o /. Exmple: Conider prmeerized curve, which r (, ) nd end (, ). The ir curve ollow he rel xi o he origin, coninue long circle in he upper hl-plne, nd inihe y ollowing he rel xi o (, ). The ngen ngle uncion i given y < / ( ) = 6π( /) / < /. π / The curve h prmeric relecion ymmery wih repec o he imginry xi. The curve doe no hve prmeric roionl ymmery. The econd curve coni o wo righ egmen nd wo circle nd

4 4 NDERS LINNÉR ( ) < /4 8π( /4) /4 < / =. 8π ( /4 ) / < /4 /4 Thi curve h prmeric roionl ymmery. Thi curve doe no hve prmeric relecion ymmery. Noe h when he curve i hough o e o poin i h exernl relecion ymmery cro he imginry xi.. Ueul mp. To e in poiion o employ mniold heoreicl ool he ollowing c re elihed. Suppoe he endpoin γ () nd γ () re ixed, hen he ngen ngle i ujec o nonliner conrin o he orm i() = γ() γ(). L e d The direcionl derivive o i() e d, in he direcion v, h he orm () () i. i v e d The pce o ngen ngle pproprie in he conex o ol qured curvure i given y he vecor pce = : [, ] i oluely coninuou, L [, ]. In he nex wo { } reul here i no need o ume hi much moohne in, u no hrm i done. Propoiion. Le W { w w( ) w( ) } conn uncion. The liner mp = = nd uppoe W i given uch h i no () Γ ( w) = w( ) e i d, Γ : W deined y i ono. Proo. Inegre y pr nd ge i() i() Γ ( w) w() e d w( ) = e d d. Given ny [,/ ] deine w W y w ( ) = ( + ( ) ) wih < w =, < nd w () =. Deine curve σ : [,/] y σ ( ) = Γ ( w ). The curve i wice dierenile wih

5 SYMMETRIZED CURVE-STRIGTENING σ i() () = e d, nd ince W ( ) σ () = e i. I ollow h σ i no conn. ume σ( ) or [,/]. I here i uncion :[,/] uch h σ() = ()( σ ) or ll [,/ ], hen i mooh σ nd i( ) e = σ ( ) = ( ) σ( ). Since i coninuou hi implie h i conn. Thi i ruled ou y umpion nd hence here re wo linerly independen vecor o he orm σ ( ) = Γ ( w ) nd σ ( ) = Γ ( w ). Propoiion. Le V { v v( ) v( ) v(/) } = + = nd uppoe V i given uch h i no conn uncion. The liner mp i ono i () Λ () v = v() e i d, i() e d, Λ : V deined y i (/) nd h -dimenionl imge pnned y ie oherwie. ic Proo. Since Λ c = e Λ or ny rel numer c, i i cceple o ume h () + ( ) =. y pliing he inervl o inegrion in hl nd uing uiuion i i een h o / i() e d = co ( ) d, Im. i() e d = Fir conider he ce Le v i() e d. V e given y v () =. Deine v < v =, < V y nd v () =. Deine curve σ : [,/] y σ ( ) = Λ ( v ). The curve i wice dierenile wih i() i(), nd σ () = e d + e d

6 6 NDERS LINNÉR i( ) i( ) i( ) i( ) = = =. σ () e e e e in i () Suppoe here i uncion : [,/ ] uch h σ () = () Λ () v or ll [,/ ], hen i mooh σ nd i() in i () = σ () = () Λ () v = () e d. Thi implie i () Re e d =. I ollow h i () e d =, conrdicion. ence here re wo linerly independen vecor o he orm Λ ( v ) nd σ ( ) = Λ ( v ). Nex conider he ce For ll v Since i () e d =. V i i rue h / i() i( u) i() v() e d = v( u) e du = ( (/) v v() ) e d, nd / / / / i() i() i() i() ve () d= v ()( e e ) d+ (/) v e d. / i() = e d = co ( ) d i ollow h () Re i ve ( ) d =. To inih he proo i uice o ind le one v V uch h i() ve () d. To hi end, noe h he curve σ, which i deined in he ir pr o he proo, move wy rom he origin unle ( ) = n π or ome ineger n. In hi ce he coninuiy o implie h i conn uncion. Thi conrdic he umpion on. I ollow h ny uch h σ( ) yield uncion v v i () () e d. V wih he propery

7 SYMMETRIZED CURVE-STRIGTENING 7.4 Sumniold o prmericlly ymmeric curve. The pce o ll geomeric curve wih + + iniil poin he origin i repreened y where denoe he poiive rel numer. + Given n elemen (,L), he correponding curve i given y i() γ () = L e d. Le Φ : e given y Φ ( )() = () ( ). Pu m = / nd le Ψ : e given y ψ( )() = () + ( ) ( m). Suppoe g i ixed complex numer nd, L > re ixed rel numer. Conider he ollowing ue: ro + i() Ω g = (, L) L e d g, ( ) = Φ =, ro i() Ω = L e d g, ( ) gl, = Φ =, rel + i() Ω = (, L) L e d g, ( m), ( ) g, = = Ψ =, nd rel i() Ω = L e d g, ( m), ( ) g,, L = = Ψ =. ll he inner produc ued in hi pper hve he propery h convergence in norm implie poin-wie convergence. Moreover, oh Φ nd Ψ re coninuou mp. Theorem : Suppoe he vecor pce i equipped wih norm uch h convergence in norm implie poin-wie convergence, nd i complee pce. Furhermore, ume h oh Φ nd Ψ re coninuou, hen he ollowing hold: ro rel + () Ωg nd Ω re cloed umniold o, nd g, ro rel () Ω nd Ω re cloed umniold o. gl, g,, L Proo: Since Φ nd Ψ re coninuou liner mp, he pce { () } cloed liner upce o. I lo ollow h { m } Φ = Φ = i, = ( ) =, Ψ ( ) = i cloed Ψ liner upce in nd =,, + i cloed liner umniold o Ψ Ψ. To del wih he ce o unconrined lengh ce () deine + φ : Φ Φ, nd φ : + y Noe h Ψ, Ψ, Ψ, i() φ (, L) = φ (, L) = L e d. Φ rel φ g, Ψ, ro Ω = () g nd Ω = () g. Recll h g φ Φ i() γ () = L e d o h γ () = g. The direcionl derivive in he direcion v = ( v v ) re given y, L

8 8 NDERS LINNÉR DφΦ(, L) v = DφΨ(, L) v = v e d + il v e d = g + i v () g v ()() γ d The ngen pce re given y T i() i() vl L L. Φ = nd T =,. ccording o he ndrd Φ Ψ, Ψ mniold heory ([] p. 97, he Sumerion Theorem) i i enough o how h he direcionl derivive i ono o prove h ro Ω = () g nd Ω rel = () g re cloed umniold. To g φ Φ φ g, Ψ, hi end, ume ir h g. I v i conn uncion o h v ( ) = R, hen + v T nd v T. For ny Φ Ψ v L TR = R he direcionl derivive i given y (, ) (, ) (, ) (, v Dφ ), ( L Φ L vl = Dφ L vl = + i) g. Ψ L Given ny c, chooe v = LRe ( c/ g) nd = Im ( c/ g). To inih uppoe h g =. The derivive impliy Ψ, L D φ (, L) v = D φ (, L) v = i v ()() γ d. Φ In he ce o Dφ Φ i uice o mimic he ep in he proo o Propoiion o how h he mp i ono, nd in he ce o D φ Ψ, ue Propoiion. The dierence in he ler ce i h he mp i ono one-dimenionl upce o, which only depend on. To inih he proo he me reoning i pplied in he ce (). Divide y L eore pplying he wo Propoiion.. SYMMETRIZING RIEMNNIN STRUCTURES +. Riemnnin rucure. The Riemnnin rucure deined on T( ) = h he ollowing orm ( v, vl),( w, wl) = v, w + v LwL. Following [4], le α, β (no oh zero) nd p [,] e given nd conider he ollowing hree dieren poiiliie or, vw, I = αv() w() + βv() w() + vwd ( ) ( ), vw, = vwd ( ) ( ) + vwd ( ) ( ), nd II vw, vpwp ( ) ( ) vwd ( ) ( ). III = +. The modiied ol qured curvure. Given rel numer ν, he generl curverighening uncionl J ν : + i given y. J ν (, L) = () d + νl L The grdien o J ν + i pir o he orm ( Fv ) T T =, where, L

9 SYMMETRIZED CURVE-STRIGTENING 9 vl = () d + ν L, nd () F( ) = d. L In [4] he generl ce wih F( x) = ( x, ( ), x ( )) d i developed. The quniie Ex( ) = x ( x, ( ), x ( )) x ( x, ( ), x ( )) d, nd W = (, x( ), x ( ) ) d x x re inroduced. Thi generl ce pecilize o he preen conex wih ( ) (), ( ), ( ) =. L In erm o he igned curvure k hi yield E = = / L = k, nd W =.. The unconrined ce nd he meric I. I i hown in [4] (nd in he ppendix) h ( ) ()( ) α + I Fx = Ex() d+ Wx β Ex() d αβ( ) + α + β. Thi pecilize o β α + ( ) IF () = ( () () ) ( () () ) L L αβ + α + β. Theorem 4. In meric I, I F preerve roion ymmerie i α =, nd relecion ymmerie i α = β. I ollow h oh ymmerie cnno e preerved imulneouly. Proo. Suppoe ( ) = ( ), hen β α( ) + ( ) IF ( ) = ( ( ) () ) ( () () ) L L αβ + α + β. I F( ) () = F( ) ( ), hen α( ) = α nd α =. Suppoe ( ) = ( ), hen () β α( ) + ( ) IF ( ) = ( ( ) () ) + L L αβ + α + β. I F( ) () = F( ) ( ), hen αβ + β = αβ + α + β. I ollow h α = β, nd hence oh ymmerie cnno e preerved ince hi implie α = β =. Remrk. ecue he meric I il o preerve oh ype o ymmerie i i excluded rom urher coniderion in hi pper.

10 NDERS LINNÉR.4 The unconrined ce nd he meric II nd III. To impliy he noion wrie h( ) = inh( ) nd ch( ) = coh( ). The generl ce o [4] re given y ch( ) ( )( ) IIFx = Ex( )ch( d ) + Wx Ex( )h( d ) h( ), nd Ex() d + Wx, p p ( )( ) IIIFx =. Ex() d + Wx ( p + ) p p They pecilize o ch( ) ()( ) IIF = k()ch( ) d k()h( ) d, nd h() ( ) III F () = ( () () p ). L Theorem. In meric II, II F preerve oh roion nd relecion ymmerie. Proo. ume h ( ) = ( ). I ollow h () = ( ) o h k ( ) = k ( ). Now ch() ( ) IIF () = k() ch( ) d k() h( ) d, nd h () ch( ) ( ) IIF ( ) = k( ) ch( ) d k( ) h( ) d h() ch( ) = kch () ( d ) kh () ( d ). h() I ollow h i he ollowing i n ideniy ch() ch( ) kch () ( d ) = kh () ( d ) h(), hen ( ) IIF ( ) IIF () = ( ) hold. Wih he help o hyperolic ideniie hi led o ch() k() ch() d h() k() h() d = ch() ch( ) h() k( ) ch( ) d ch() k( ) h( ) d. h() Collec erm nd ge h() ch( ) k() ch() d = ( h() h( ) ch() ch( ) + ch() ch( ) ) k( ) h( ) d. One more pplicion o hyperolic ideniy produce, h() k( ) ch( ) d = ( ch() ) k( ) h( ) d

11 SYMMETRIZED CURVE-STRIGTENING which conver o k () h () d = ch () k () h () d h () k () ch () d = k () h ( ) d. uiuion how h hi i indeed n ideniy when k ( ) = k( ) hold. The ce () = ( ) i imilr nd he proo i omied here or he ke o reviy. Theorem 6. In meric III, III F preerve oh roion ymmerie nd ymmerie o relecion i nd only i p = /. In meric III, III F lwy preerve roion ymmerie. Proo. Since ( ) III F () = ( () ( p )), L i ollow h ( ) = ( ) implie ( )() ( ) IIIF = IIIF ( ). I ( ) = ( ), hen (/ ) =, nd ( )() ( ) IIIF + IIIF ( ) = ( p)/ L. ence ( )() ( ) IIIF = IIIF ( ) or ll uch h ( ) = ( ) i nd only i p = /.. Funcionl in he conrined ce. In he unconrined ce nohing i umed ou he endpoin. Suppoe ined (, ) i given nd i() (, ) = γ() = γ() γ() = L e d. I i ueul o inroduce wo rel-vlued uncionl G, : G + given y G (, L) = L co ( ) d, G ( ), L = L in ( ) d. Thi ime (, ( ), ( ) ) = Lco ( ), (, ( ), ( ) ) = Lin ( ), nd ( ) = Lin, ( ) co = L, ( ) ( ) = =. I ollow h E () = L in () d y() =, W =, nd E () = L co () d = x(), W =..6 The meric II nd he uncionl in he conrined ce. Theorem 7. In meric II, II G preerve roionl ymmerie well ymmerie o relecion, nd IIG preerve roionl ymmerie. Proo. The wo ce re imilr o IIG i le o he reder. The componen i given y

12 NDERS LINNÉR ch() G ( )() () ( ) () ( ) = y ch d y h d + h () Suppoe ( ) = ( ) o h y( ) = L in ( ) d = L in ( ) d = y( ).. I ollow h G ( ) () G ( ) ( ) = ch() ch( ). ych ( ) ( d ) ch ( d ) yh ( ) ( d ) + h() uiuion nd ome lger how h. + yh () ( d ) = ch() yhd () () Fir collec he erm conining nd evlue he remining inegrl ch { () ch ( ) ( ( ) ( ))} { () h + ch ch = ch( ) h() h() }. The re o he erm impliy ollow ch() ch( ) ych () ( d ) + yhd () () = h() ch() ch() ch() + h() h() ch() y() ch() d h() y() h() d + y() h() d = h() ch() y() { h() ch() ch() h() } d + ch() y() h() d h() = ch() ( ch() ) y (){ h () h( ) } d ch () h() + = h() inl conolidion o he erm prove h G ( ) () ( )( ) = G. Nex, uppoe h ( ) = ( ) o h / in ( d ) = in ( udu ) = in ( d ), / / nd hence =. Conclude h. y( ) = L in ( ) d = L in ( ) d = y( ) = y( ) I ollow h G ( ) () + G ( ) ( ) = pir o uiuion yield ch() + ch( ) y() ch( ) d + y() ch( ) d y() h( ) d h()

13 SYMMETRIZED CURVE-STRIGTENING ch() + ch( ) G ( ) () + G ( ) ( ) = ych () ( d ) yhd () () h() Sndrd hyperolic ideniie nd uiuion how h G ( ) () ( )( ) + G =. ch() + ch() ch() h() h() ch() y() ch() d h() y() h() d y() h() d = h() ch() ch() y (){ h() ch () ch() h ()} d yh () ( d ) h() = h() G () ( ) G I ollow h ( ) = ( )..7 The meric II nd ymmerized curve-righening. The uncionl J ν i deined on ll curve wih iniil poin he origin. The grdien vecor ield J ν i ecion o he ngen undle o he pce o ll curve. There i projecion o hi grdien vecor ield ono he vriou ngen undle ocied wih endpoin conrin nd poile lengh conrin. The low induced y he projeced vecor ield i reerred o curve-righening low. i hown in [4], he projeced vecor ield in he ngen undle o conrined curve i no in generl ngen o he pce o ymmeric curve. Moreover, he pce o ymmeric curve hve norml undle wih ininie dimenionl ier nd correponding non-rivil projecion ono he ngen undle o ymmeric curve. Thee diiculie re circumvened y he ollowing reul. Theorem 8. The curve-righening low in meric II preerve oh roion ymmerie nd ymmerie o relecion. ν + Proo. Wrie J = ( F, v L ) T T =. Denoe he projecion o F ono he pce o curve wih conrined endpoin y π F. There re wo rel clr ield λ nd λ uch h F π = F + λ G + λ G. The ngency condiion re given y DG ( π F ) = nd DG ( ) π F =. Since ll he vecor ield eore he projecion preerve roionl ymmerie, roionl ymmerie re lo preerved er he projecion. In he ce o ymmerie o relecion hing re more compliced. The roule i creed y he ilure o G o preerve ymmerie o relecion. Conider he ollowing liner yem DG ( ) G ( ) DG ( ) G ( ) ( ) ( ) λ DG F DG ( ) ( ) ( ) ( ) ( ) ( ) G DG G λ =. DG F I ollow h λ ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = DG G DG F DG G DG F )/de. Since ( ) = ( ) i ollow h ch() F( )() = k() ch( ) d k() h( ) d. h () Recll h = o h

14 4 NDERS LINNÉR ch() G ( )() () ( ) () ( ) = y ch d y h d. h () lo noe h hving = elimine he erm in DG correponding o vriion o he lengh. Now DG ( ) ( ) F = co ( ) ( ) d where ch() () = kch () ( d ) kh () ( d ). h () Suiuion nd hyperolic ideniie yield ch() + ch( ) () + ( ) = kch () ( d ) kh () ( d ) h() ch() + ch() ch() h() h() = ch() k() ch() d h() k() h() d k() h( ) d h(). = ch() k() ch() d ch() ch() ch() h() k() ch() d ch() k( hd ) ( ) + () h ch() ch() = khd () () kh () ( d ) = h() I ollow h DG ( ) ( ) F = co ( ) ( ) d =. Uing he me exc reoning i ollow h DG ( ) ( ) G = co ( ) ( ) g d =, where ch() g () = ych () ( d ) yh () ( d ), h () nd y ( ) h he me ymmery properie k ( ). Wih hee wo reul i ollow h λ =..8 The meric III nd he uncionl in he conrined ce. Theorem 9. I p = / in he meric III, hen IIIG preerve roionl ymmerie well ymmerie o relecion nd III G preerve roionl ymmerie u no ymmerie o relecion. Proo. The componen re given y

15 SYMMETRIZED CURVE-STRIGTENING yd () / / G ( )() =, yd () ( + /) / / xd () + / / G ( ) () =. xd () + ( + /) / / Suppoe ( ) = ( ) o h y( ) = L in ( ) d = L in ( ) d = y( ) I ollow h nd, x( ) = L co ( ) d = L co ( ) d = x( ), y() d = y( u) du = ( / ) + y() d, nd / / / x() d = x( u) du = ( / ) + x() d. / / / yd () ( + /) / / G ( )( ) =, yd ( ) ( /) ( / ) / / xd () + ( + /) / / G ( ) ( ) =. xd ( ) + ( /) + ( / ) / / One more impliicion prove h G ( ) () ( )( ) = G nd G ( )() ( ) = G ( ). Nex, uppoe h ( ) = ( ). in he proo o Theorem 7 conclude h = nd h y( ) = y( ). Wih y() d = y( u) du = ( / ) y() d = y() d / / / / G () ( ) = G. Similrly ( ) / G ( ) ( ) i ollow h ( ) ( ) xd () + / + / =, xd ( ) + ( /) + ( / ) / / nd hence G ( ) () = G ( ) ( ).

16 6 NDERS LINNÉR.9 The meric III nd ymmerized curve-righening. The reoning i in hi ce imilr o he ce o he meric II. Theorem. The curve-righening low wih p = / in meric III preerve oh roion ymmerie nd ymmerie o relecion. Proo. Ue he me noion in he proo o Theorem 8. Since ll he vecor ield eore he projecion preerve roionl ymmerie, roionl ymmerie re lo preerved er he projecion. In he ce o ymmerie o relecion hing re more compliced. The roule i gin creed y he ilure o G o preerve ymmerie o relecion. Recll h = nd λ ( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = DG G DG F DG G DG F )/de. Since ( ) = ( ) implie (/ ) = i mu e h F( )() = ()/ L. I i lo rue h G ( )() = y() d. / Now DG ( ) ( ) F = ()co () d = ecue / ()co () d = ()co () d. / I i lo rue h DG ( ) ( ) G co ( ) ( ) L y d = d = / ecue uiuion how h / L co ( ) y( ) d d L co ( ) y( ) d = d. / / / Wih hee wo reul i ollow h λ =.. Exmple. In ll exmple he lengh i unconrined nd ν = o here i no lengh penly. The igure depic lowime evoluion y providing npho long he negive grdien rjecory. The imeinervl eween conecuive curve i no conn or even monoone. In ll ce he iniil curve h very high elic energy. Thi produce rpid chnge long he low. To cpure hi iny inervl i ued he r o he low. In Figure nd 6 he imeinervl i lo horened coniderly ler he curve gin exhii igniicn chnge. In he ir hree igure he iniil curve coni o righ egmen ollowed y clockwie circle nd inihing righ egmen. The Figure illure how he rdiionl curve righening low rek ymmerie. Thi igure milirize he reder wih hreedimenionl repreenion o he evoluion in ime. In Figure meric III i ued wih p =, in

17 SYMMETRIZED CURVE-STRIGTENING 7 Figure meric III i ued wih p = /, nd in Figure meric II i ued. In oh o he l wo ce he ymmery i preerved. Figure 4 ue dieren iniil curve o illure more clerly he geomeric dierence eween meric III wih p = / nd meric II. The iniil curve i in oh ce he me iny pirl wih no ymmerie. Figure illure pinning in he rdiionl curve righening low. The iniil curve i given y hree loop o circulr igure eigh. oh meric II nd meric III wih p = / void pinning. The low in meric III i hown in Figure 6. ll he exmple conidered converge o one loop o he Euler igure eigh. curren opic o inveigion i o deermine excly which o he criicl poin ininiy re limi o negive grdien rjecorie. The doule covered igure eigh h emerged igniicn e ce ince i expoe lck o longerm iliy in curren numericl implemenion. 4. PPENDIX promied, hi ppendix upplie he proo o he generl grdien ormul derived in [4]. Given ny coninuouly dierenile uncionl F : R, he direcionl derivive DF( x) : R nd he grdien Fx ( ) iy Fx ( ), v = DFxv ( ). ere, i he inner produc in. Theorem. Le : R R e coninuouly dierenile. Denoe he pril derivive wih repec o he econd nd hird rgumen y x, nd x. Conider only uncion x ( x, (), x () ), ( x, (), x () ), ( x, (), x ()) L [, ]. Pu x x Ex( ) = x ( x, ( ), x ( )) x ( x, ( ), x ( )) d, nd Wx = x (, x( ), x ( ) ) d. The ollowing i rue or Fx ( ) = ( x, ( ), x ( )) d (I) The grdien wih repec o wv, I = αwv ()() + βwv ()() + wvd ()() i given y α( ) + IFx ( ) = Ex( d ) + Wx β Ex( d ) αβ( ) α β + + (II) The grdien wih repec o i given y. wv, = wvd ()() + wvd ()() II uch h

18 8 NDERS LINNÉR ch( ) IIFx ( ) = Ex( )ch( d ) + Wx Ex( )h( d ) h( ) (III) The grdien wih repec o i given y. wv, wpvp ( )( ) wvd ()() III = + Ex() d + Wx, p p IIIFx ( ) =. Ex() d + Wx ( p + ) p p Proo. I uice o veriy he relion Fx ( ), v = DFxv ( ) in he hree ce. Fir, ndrd compuion uing inegrion y pr how h DFxv ( ) = ( ( x, ( ), x ( )) v ( ) + ( x, ( ), x ( )) v ( )) d= v ( ) ( x, ( ), x ( )) d+ x x x x ( x, ( ), x ( )) x ( x, ( ), x ( )) d vd ( ) = Wx v ( ) + Ex( vd ) ( ). I he meric i, I, hen α I Fx ( ), v = Wx β Ex( d ) v ( ) + I αβ( ) + α + β α( ) + β Ex( ) d Wx β Ex( ) d + v( ) + αβ( ) + α + β α Ex() Wx β Ex() d + v() d. αβ( ) + α + β The l inegrl impliie o α Ex()() v d Wx β Ex() d + ( v() v( ) ). αβ( ) + α + β The erm wih he cor v ( ) dipper. Collec he erm wih he cor v ( ) nd rerrnge o ee h v ( ) i muliplied y I ollow h αβ ( ) + β αβ β Ex () d + αβ( ) + α + β αβ( ) + α + β αβ( ) β α + + Wx = Wx. αβ( ) + α + β αβ( ) + α + β IFx ( ), v = Wv x ( ) + Ex( vd ) ( ) = DFxv ( ) I. When he meric i, II, he ir min erm o he econd inegrl i rewrien wih he help o

19 SYMMETRIZED CURVE-STRIGTENING 9 Leiniz rule ch( ) IIF( x), v = Ex( )ch( ) d Wx Ex( )h( ) d v( ) d II + + h( ) Ex( ) Ex( )h( ) d h( ) + + Wx Ex( )h( ) d vd (). h( ) Inegre he econd min erm o he econd inegrl y pr h( ) Wx Ex()h( ) d dv() = h( ) ch( ) Wx Ex()h( ) d v() Wx Ex()h( ) d v() d. h( ) lo inegre he ir min erm o he ir inegrl y pr gin uing Leiniz rule Ex()ch( ) d v() d = v() d Ex()h( ) d = v( ) Ex( )h( ) d Ex( )h( ) d v( ) d Once oh o hee compuion re comined, i ollow h Finlly, IIFx ( ), v = Wx v ( ) + Ex( vd ) ( ) = DFxv ( ) II. ( ) F(), x v = W v( p) + E ()() v d + E () + W v () d III III = W v() + E ()() v d p x x x x p x x

20 NDERS LINNÉR REFERENCES [] Linnér,., Some properie o he curve righening low in he plne, Trn. mer. Mh. Soc. 4 (989), 6-67 [] Linnér,., Seepe decen ued ool o ind criicl poin o k deined on curve in he plne wih rirry oundry condiion, Geomeric nlyi nd Compuer Grphic, MSRI Pulicion, Vol. 7, Springer-Verlg, 99, 7-8 [] Linnér,., Curve-righening nd he Pli-Smle condiion, Trn. mer. Mh. Soc. (998), [4] Linnér,., Grdien, preerred meric nd ymmerie, Preprin () [] rhm R., Mrden J. E. nd Riu T., Mniold, Tenor nlyi, nd pplicion, nd Ed. 988 Springer-Verlg

21 Symmerized.n Figure. Unconrined lengh uing no lengh penly in Meric III wih p

22 Symmerized.n Figure. Unconrined lengh uing no lengh penly in Meric III wih p

23 Symmerized.n Figure. Unconrined lengh uing no lengh penly in Meric II Figure 4. Compre meric III wih p nd meric II

24 Symmerized.n

25 Symmerized.n Figure. Meric III wih p illuring pinning

26 Symmerized.n 6 Figure 6. Meric III wih p illuring removl o pinning