On the H^1(ds)-gradient flow for the length functional
OON THE H ( ds ) -GRADIENT FLOW FOR THE LENGTH FUNCTIONAL PHILIP SCHRADER, GLEN WHEELER, AND VALENTINA-MIRA WHEELER
Abstract.
In this article we consider the length functional defined on the space ofimmersed planar curves. The L ( ds ) Riemannian metric gives rise to the curve shorteningflow as the gradient flow of the length functional. Motivated by the vanishing of the L ( ds ) Riemannian distance, we consider the gradient flow of the length functional withrespect to the H ( ds )-metric. Circles with radius r shrink with r ( t ) = (cid:112) W ( e c − t )under the flow, where W is the Lambert W function and c = r + log r . We conducta thorough study of this flow, giving existence of eternal solutions and convergence forgeneral initial data, preservation of regularity in various spaces, qualitative properties ofthe flow after an appropriate rescaling, and numerical simulations. Contents
1. Introduction 21.1. The gradient flow for length in (Imm , L ( ds )) 21.2. Vanishing Riemannian distance in (Imm , L ( ds )) 31.3. The gradient flow for length in (Imm , H ( ds )) 32. Metrics on spaces of immersed curves 62.1. Vanishing Riemannian distance in (Imm , L ( ds )) 73. Symmetries of metrics and gradient flows 94. The gradient flow for length with respect to the H ( ds ) Riemannian metric 114.1. Derivation, stationary solutions and circles 114.2. Existence and uniqueness 144.3. Convergence 215. Shape evolution and asymptotics 255.1. Generic qualitative behaviour of the flow 255.2. Remarks on the numerical simulations 265.3. Evolution and convergence of an exponential rescaling 265.4. The H ( ds )-flow in Imm k spaces 295.5. Curvature bound for the rescaled flow 315.6. Isoperimetric deficit 325.7. A chord-length estimate and embeddedness 34References 36 Mathematics Subject Classification. a r X i v : . [ m a t h . DG ] M a r P. SCHRADER, G. WHEELER, AND V.-M. WHEELER Introduction
Consider the length functional:(1) L ( γ ) := (cid:90) S (cid:12)(cid:12) γ (cid:48) (cid:12)(cid:12) du defined on Imm , the space of (once) differentiable curves γ : S → R with | γ (cid:48) | (cid:54) = 0.The definition of a gradient of L requires a notion of direction on Imm , that is an innerproduct or more generally a Riemannian metric (cid:104)· , ·(cid:105) . The gradient is then characterisedby d L = (cid:104) grad L , ·(cid:105) .To calculate the (Gateaux) derivative d L take a variation γ : ( − ε, ε ) × S → R , ∂ ε γ | ε =0 = V and calculate d L γ V = (cid:90) S (cid:104) ∂ ε ∂ u γ, ∂ u γ (cid:105)| ∂ u γ | du (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = − (cid:90) (cid:104) T s , V (cid:105) ds = − (cid:90) k (cid:104) N, V (cid:105) ds (2)Here the inner product is the Euclidean one, u is the given parameter along γ = ( x, y ), s is the Euclidean arc-length parameter, T = ( x s , y s ) is the unit tangent vector, k thecurvature scalar, and N = ( − y s , x s ) is the normal vector.As for the inner product or Riemannian metric in Imm we might choose either of (cid:104) v, w (cid:105) L := (cid:90) S (cid:104) v, w (cid:105) du , or (cid:104) v, w (cid:105) L ( ds ) := (cid:90) S (cid:104) v, w (cid:105) ds for v, w vector fields along γ . The former is simpler, but from the point of view of geometricanalysis (and in particular geometric flows) the latter is preferable because it is invariantunder reparametrisation of γ , and this invariance carries through to the correspondinggradient flow (see Section 3). Note that the L ( ds ) product is in fact a Riemannian metricbecause it depends on the base point γ through the measure ds .1.1. The gradient flow for length in (Imm , L ( ds )) . Indeed (2) shows that the gra-dient flow of length in the L ( ds ) metric is the famous curve shortening flow proposed byGage-Hamilton [12]:(3) X t = X ss = κ = kN where X : S × (0 , T ) → R is a one-parameter family of immersed regular closed curves, X ( u, t ) = ( x ( u, t ) , y ( u, t )) and s, T, k, N are as above.The curve shortening flow moves each point along a curve in the direction of the cur-vature vector at that point. Concerning local and global behaviour of the flow, we have: Theorem 1.1 (Angenent [4], Grayson [14], Gage-Hamilton [12], Ecker-Huisken [10]) . Consider a locally Lipschitz embedded curve X . There exists a curve shortening flow X : S × (0 , T ) → R such that X ( · , t ) (cid:38) X in the C / -topology. The maximal time ofsmooth existence for the flow is finite, and as t (cid:37) T , X ( · , t ) shrinks to a point { p } . Thenormalised flow with length or area fixed exists for all time. It becomes eventually convex,and converges exponentially fast to a standard round circle in the smooth topology.Remark . In the theorem above, we make the following attributions. Angenent [4]showed that the curve shortening flow exists with locally Lipschitz data where convergenceas t (cid:38) C / -topology. Gage-Hamilton [12] showed that a convex curve contracts to around point, whereas Grayson [14] proved that any embedded curve becomes eventuallyconvex. There are a number of ways that this can be proved; for instance we also mentionHuisken’s distance comparison [16] and the novel optimal curvature estimate method in[1]. N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 3 The curve shortening flow has been extensively studied and found many applications.We refer the interested reader to the recent book [2].1.2.
Vanishing Riemannian distance in (Imm , L ( ds )) . Every Riemannian metricinduces a distance function defined as the infimum of lengths of paths joining two points.For finite dimensional manifolds the resulting path-metric space has the same topologyas the manifold, but for infinite-dimensional manifolds it is possible that the path-metricspace topology is weaker (so-called weak Riemannian metrics ). Furthermore, as famouslydemonstrated by Michor-Mumford in [21], it is possible that the Riemannian distance isactually trivial.The example given by Michor-Mumford is the space (Imm , L ( ds )) quotient by thediffeomorphism group Diff( S ) of S . We call this space Q = Imm / Diff( S ). WhileImm is an open subset of C ( S , R ) and so (Imm , L ( ds )) is a Riemannian manifold,the action of Diff( S ) is not free (see [21, Sections 2.4 and 2.5]), and so the quotient Q isnot a manifold but is an orbifold. Theorem 2.1 (Michor-Mumford [21]) . The Riemannian distance in ( Q , L ( ds )) is trivial. This surprising fact is shown by an explicit construction in [21] of a path between orbitswith arbitrarily small L ( ds )-length, which for the benefit of the reader we briefly recallin Section 2.1. A natural question arising from Michor-Mumford’s work is if the inducedmetric topology on the Riemannian manifold (Imm , L ( ds )) is also trivial. This wasconfirmed in [5] as a special case of a more general result. Theorem 2.2.
The Riemannian distance in (Imm , L ( ds )) is trivial. Here we give a different proof of Theorem 2.2 using a detour through small curves. Thesetup and proof is given in detail in Section 2.1.We can see from (2) that the curve shortening flow (3) is indeed the L ( ds )-gradientflow of the length functional in Imm , not the quotient Q . Theorem 2.2 yields that theunderlying metric space that the curve shortening flow is defined upon is trivial , andtherefore this background metric space structure is useless in the analysis of the flow.While it could conceivably be true that the triviality of the Riemannian metric topologyon Imm , L ( ds ) is important for the validity of Theorem 1.1 and the other nice propertiesthat the curve shortening flow enjoys, one naturally wonders if this is in fact the case: Whatdo gradient flows of length look like on Imm , with other choices of Riemannian metric?1.3. The gradient flow for length in (Imm , H ( ds )) . We wish to choose a metric that(a) yields a non-trivial Riemannian distance; and (b) produces a non-trivial gradient flow.One way of doing this (similar to that described by Michor-Mumford [21]) is to view the L ( ds ) Riemannian metric as an element on the Sobolev scale of metrics (as the H ( ds )metric). The next most simple choice is therefore the H ( ds ) metric:(4) (cid:104) v, w (cid:105) H ( ds ) := (cid:104) v, w (cid:105) L ( ds ) + (cid:104) v s , w s (cid:105) L ( ds ) Note that we have set the parameter A from [21, Section 3.2, Equation (5)] to 1 and weare considering the full space, not the quotient. In contrast to the L ( ds ) case, the H ( ds )distance is non-trivial [21]. Diff( S ) is the regular Lie group of all diffeomorphisms φ : S → S with connected components Diff + ( S ),Diff − ( S ) given by orientation preserving and orientation reversing diffeomorphisms respectively. P. SCHRADER, G. WHEELER, AND V.-M. WHEELER
Remark . There is an expanding literature on the multitude of alternative metricsproposed for quantitative comparison of shapes in imaging applications (see for example[6, 7, 11, 17, 19, 21, 23, 27, 28, 29, 30, 31, 32]). It might be interesting to comparethe dynamical properties of the gradient flows of length on Imm with respect to otherRiemannian metrics. We note also that the study of Sobolev type gradients is far fromnew. We mention the comprehensive book on the topic by Neuberger [24], and the flowstudied in [28] for applications to active contours is closely related to the one we studyhere. A recurring theme seems to be better numerical stability for the Sobolev gradientcompared to its L counterpart. However, in this article we focus on analytical aspects.The steepest descent H ( ds )-gradient flow for length (called the H ( ds ) curve shorten-ing flow ) on maps in H ( S , R ) is a one-parameter family of maps X : S × I → R ( I an interval containing zero) where for each t , X ( · , t ) ∈ H ( S , R ) and(5) ∂ t X ( s, t ) = − (cid:16) grad H ( ds ) L X ( · ,t ) (cid:17) ( s ) = − X ( s, t ) − (cid:90) L X (˜ s, t ) G ( s, ˜ s ) d ˜ s where G is given by G ( s, ˜ s ) = cosh (cid:0) | s − ˜ s | − L (cid:1) − L ) for 0 ≤ s, ˜ s ≤ L . Our derivation of this is contained in Section 4.1.An instructive example of the flow’s behaviour is exhibited by taking any standardround circle as initial data. A circle will shrink self-similarly to a point under the flow,taking infinite time to do so. The circle solutions can be extended uniquely and indefinitelyin negative time as well, that is, they are eternal solutions (see also Section 4.1).We set C to be the space of constant maps. While (5) does not make sense on C , ourfirst main result is that everywhere else on H ( S , R ) it does, and we are able to obtaineternal solutions for any initial data X ∈ H ( S , R ) \ C . This is Theorem 4.12, which isthe main result of Sections 4.2–4.2.2: Theorem 4.12.
For each X ∈ H ( S , R ) \ C there exists a unique eternal H ( ds ) curveshortening flow X : S × R → R in C ( R ; H ( S , R ) \ C ) such that X ( · ,
0) = X . In Section 4.3 we study convergence for the flow, showing that the flow is asymptoticto a constant map in C . Theorem 4.18.
Let X be an H ( ds ) curve shortening flow. Then X converges as t → ∞ in H to a constant map X ∞ ∈ H ( S , R ) . Numerical simulations of the flow show fascinating qualitative behaviour for solutions.Figures 1 and 2 exhibit three important properties: first, that there is no smoothing effect- it appears to be possible for corners to persist throughout the flow. Second, that theevolution of a given initial curve is highly dependent upon its size, to the extent that asimple rescaling dramatically alters the the amount of re-shaping along the flow. Third,the numerical simulations in Figures 1 and 2 indicate that the flow does not uniformlymove curves closer to circles. The scale-invariant isoperimetric ratio I is plotted alongsidethe evolutions and for an embedded barbell it is not monotone . We have given somecomments on our numerical scheme and a link to the code in Section 5.2.Despite the lack of a generic smoothing effect, what we might hope is that a generic preservation effect holds. In Section 5.4 we consider this question in the C k regularityspaces (here k ∈ N ), and show that this regularity is indeed preserved by the flow. We This is also the case for the classical curve shortening flow, as explained in [13].
N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 5 Figure 1.
Left: (a),(b),(c) initial side lengths 1,2,4 respectively, time step0 .
2, every 5th out of 50 steps shown. Right: evolution of the isoperimetricratio.consider the question of embeddedness in Section 5.7, with the main result there showingthat embedded curves with small length relative to their chord-arclength ratio will remainembedded. Since the chord-arclength ratio is scale-invariant but length is not, we commentthat this condition can always be satisfied by rescaling the initial data.We summarise this in the following theorem.
Theorem 1.4.
Let k ∈ N be a non-negative integer. For each X ∈ Imm k there exists aunique eternal H ( ds ) curve shortening flow X : S × R → R in C ( R ; Imm k ) such that X ( · ,
0) = X .Furthermore, suppose X satisfies (6) inf s ∈ [0 , L ] C h ( s ) S ( s ) > L (cid:112) (cid:107) X (cid:107) ∞ e L √ (cid:107) X (cid:107) ∞ . where C h and S are the chord and arclengths respectively. Then X ( t ) is a family of em-beddings. Although the H ( ds ) curve shortening flow disappears (in infinite time), we are in-terested in identifying if it asymptotically approaches any particular shape. In orderto do this, we define the asymptotic profile of a given H ( ds ) curve shortening flow Y : S × R → R by Y ( t, u ) := e t ( X ( t, u ) − X ( t, . Because of the exponential rescaling bounds for Y and its gradient become more difficultthan for X . On the other hand, scale invariant estimates for X such as the chord-arc ratioand isoperimetric ratio carry through directly to estimates on Y . Furthermore, for H ( ds ) P. SCHRADER, G. WHEELER, AND V.-M. WHEELER
Figure 2.
Evolution for a barbell inital curve. 70 steps of size 0.1.curve shortening flows on the C space the curvature scalar k is well defined, and we canask meaningfully if curvature remains controlled along Y (on X , it will always blow up).By considering the asymptotic profile we hope to be able to identify limiting profiles Y ∞ for the flow. For the vanilla flow X , the limit is always a constant map. In stark contrastto this, possible limits for Y are manifold. We are able to show that the asymptoticprofile Y does converge to a unique limit Y ∞ depending on the initial data X , but itseems difficult to classify precisely what these Y ∞ look like. For the curvature, we showin Section 5.5 that it is uniformly bounded for C initial data, and that the profile limit Y ∞ is immersed with well-defined curvature (Theorem 5.7). On embeddedness, the sameresult as for X applies due to scale-invariance. The isoperimetric deficit D Y on Y (in ascale-invariant sense) is studied in Section 5.6. It isn’t true that the deficit is monotone,or improving, but at least we can show that the eventual deficit of the asymptotic profilelimit Y ∞ is bounded by a constant times the deficit of the initial data X ; this is sharp.We summarise these results in the following theorem. Theorem 1.5.
Let k ∈ N be a non-negative integer. Set B to H ( S , R ) \ C for k = 0 and otherwise set B to C k ( S , R ) \ C . For each X ∈ B there exists a non-trivial Y ∞ ∈ H ( S , R ) \ C such that the asymptotic profile Y ( t ) → Y ∞ in C as t → ∞ .Furthermore: • Y ∞ is embedded if at any t ∈ (0 , ∞ ) the condition (6) was satisfied for X ( t ) • If k ≥ , and X (0) is immersed, then Y ∞ is immersed with bounded curvature • There is a constant c = c ( (cid:107) X (0) (cid:107) ∞ ) such that the isoperimetric deficit of Y ∞ satisfies D Y ∞ ≤ c D X (0) . Acknowledgements
The first author is grateful to Shinya Okabe and Kazumasa Fujiwara for helpful con-versations. 2.
Metrics on spaces of immersed curves
Let C k ( S , R ) be the usual Banach space of maps with continuous derivatives up toorder k . Our convention is that | S | = | [0 , | = 1. For 1 ≤ k ≤ ∞ we defineImm k := { γ ∈ C k ( S , R ) : (cid:12)(cid:12) γ (cid:48) ( u ) (cid:12)(cid:12) (cid:54) = 0 } . N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 7 Note that Imm k is an open subset of C k ( S , R ).The tangent space T γ Imm k ∼ = C k ( S , R ) consists of vector fields along γ . We definethe following Riemannian metrics on Imm for v, w ∈ T γ Imm : (cid:104) v, w (cid:105) L := (cid:90) (cid:104) v, w (cid:105) du (cid:104) v, w (cid:105) H := (cid:104) v, w (cid:105) L + (cid:10) v (cid:48) , w (cid:48) (cid:11) L (cid:104) v, w (cid:105) L ( ds ) := (cid:90) L ( γ )0 (cid:104) v, w (cid:105) ds (cid:104) v, w (cid:105) H ( ds ) := (cid:104) v, w (cid:105) L ( ds ) + (cid:104) v s , w s (cid:105) L ( ds ) (7)The length function (1) is of course well-defined on the larger Sobolev space H ( S , R ),as are the L , H and L ( ds ) products above. However, the H ( ds ) product is not well-defined because of the arc-length derivatives, even if one restricts to curves which arealmost everywhere immersed.We remark that the L ( ds ) metric on Imm is an example of a weak Riemannian metric -a purely infinite dimensional phenomenon where the topology induced by the Riemannianmetric is weaker than the manifold topology. In fact, even for a strong Riemannian metric,geodesic and metric completeness are not equivalent (as guaranteed by the Hopf-Rinowtheorem in finite dimensions) and it is not always the case that points can be joined byminimising geodesics (for an overview of these and related facts see [9]).2.1.
Vanishing Riemannian distance in (Imm , L ( ds )) . Consider curves γ , γ ∈ Imm and a smooth path α : [0 , → Imm with α (0) = γ and α (1) = γ . The L ( ds )-length of this smooth path is well-defined and given by(8) L L ( ds ) ( α ) := (cid:90) (cid:107) α (cid:48) ( t ) (cid:107) L ( ds ) dt = (cid:90) (cid:32)(cid:90) L ( α ( t ))0 | α t ( t, s ) | ds (cid:33) dt . As usual, one defines a distance function associated with the Riemannian metric byd L ( ds ) ( γ , γ ) := inf {L L ( ds ) ( α ) : α piecewise smooth path from γ to γ } Since the L ( ds ) metric is invariant under the action of Diff( S ) it induces a Riemannianmetric on the quotient space Q (except at the singularities) as follows. Let π : Imm → Q be the projection. Given v, w ∈ T [ γ ] Q choose any V, W ∈ T γ Imm such that π ( γ ) = [ γ ] ,T γ π ( V ) = v, T γ π ( W ) = w . Then the quotient metric is given by (cid:104) v, w (cid:105) [ γ ] := (cid:68) V ⊥ , W ⊥ (cid:69) γ,L ( ds ) where V ⊥ and W ⊥ are projections onto the subspace of T γ Imm consisting of vectorswhich are tangent to the orbits. This is just the space of vector fields along γ in thedirection of the normal N to γ , and so V ⊥ = (cid:104) V, N (cid:105) N . The length of a path π ( α ) in Q according to the quotient metric is then L Q L ( ds ) ( π ( α )) = (cid:90) (cid:18)(cid:90) L (cid:12)(cid:12)(cid:12) α ⊥ t (cid:12)(cid:12)(cid:12) ds (cid:19) dt and the distance isd Q L ( ds ) ([ γ ] , [ γ ]) = inf {L Q L ( ds ) ( π ( α )) : π ( α ) piecewise smooth path from [ γ ] to [ γ ] } This is the distance function that Michor and Mumford have shown to be identically zero(Theorem 2.1). They also point out (cf. [21, Section 2.5]) that for any smooth path α between curves γ , γ , there exists a smooth t -dependent family of reparametrisations P. SCHRADER, G. WHEELER, AND V.-M. WHEELER p (0) p (cid:0) (cid:1) p (1) p n (cid:0) (cid:1) p n (cid:0) (cid:1) p n (cid:0) (cid:1) p n ( t, θ i ) Figure 3.
The distortion of a path p to one that is shorter, according tothe L ( ds )-metric. For the sake of clarity, only one side of the curves onthe shorter path p n is pictured. φ : [0 , → Diff( S ) such that the reparametrised path ˜ α ( t, u ) := α ( t, φ ( t, u )) has pathderivative ˜ α t ( t ) which is normal to ˜ α ( t ) . Thus an equivalent definition isd Q L ( ds ) ([ γ ] , [ γ ]) = inf {L L ( ds ) ( α ) : α p.w. smooth with α (0) ∈ [ γ ] , α (1) ∈ [ γ ] } Theorem 2.1 (Michor-Mumford [21]) . For any ε > and [ γ ] , [ γ ] in the same pathcomponent of Q there is a path α : [0 , → Imm ∞ satisfying α (0) ∈ [ γ ] , α (1) ∈ [ γ ] andhaving length L L ( ds ) ( α ) < ε . Since it is quite a surprising result and an elegant construction, we include a descriptionof the proof. The idea is to show that we may deform any path α in Imm ∞ to a new path α n that remains smooth but has small normal projection, and whose endpoint changesonly by reparametrisation.So, let us consider a smooth path α : [0 , → Imm ∞ such that α (0) = γ and α (1) = γ . We choose evenly-spaced points θ , . . . , θ n in S and move γ ( θ i ), via α (2 t ), to theireventual destination γ ( θ i ) twice as fast. The in-between points ψ i = ( θ i − + θ i ) / θ i are at their destination, the points γ ( ψ i ) may begin to move via α . They should alsomove twice as fast as before. A graphical representation of this is given in Figure 3.The resultant path α n has small normal projection (depending on n ) but also longerlength (again depending on n ). The key estimate in [21, Section 3.10] shows that thelength of the path α n increases proportional to n and the normal projection decreasesproportional to n . Since the normal projection is squared, this means that the length of α n is proportional to the length of α times n .In other words, for any ε > γ to a reparametrisation of γ by a pathwith L ( ds )-length less than ε , and so the distance between them is zero. Note that as paths in the full space Imm , ˜ α is different to α , but they project to the same path in Q . We remark that this phenomenon of triviality of the metric topology induced by the Riemannian L ( ds )metric on the quotient space is also established in higher dimensions, see [20]. N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 9 Note that the corresponding result does not immediately follow for the full space Imm because in the full space the tangential component of the path derivative is also measured.Indeed, we could apply the Michor-Mumford construction to obtain a path α n whose de-rivative has small normal component, and then introduce a time-dependent reparametrisa-tion to set the tangential component to zero, but of course the reparametrisation changesthe endpoint to a reparametrisation of the original endpoint. However, as we show inthe following theorem, it is still possible to get the desired result by diverting through asufficiently small curve. Theorem 2.2.
The L ( ds ) -distance between any two curves in the same path componentof Imm vanishes.Proof. Let γ , γ ∈ C ∞ ( S , R ) be smooth immersions in the same path component, andlet x be another curve in the same component with (cid:107) x (cid:48) (cid:107) L ∞ < (cid:0) ε (cid:1) / (for example, x couldbe a sufficiently small scalar multiple of γ ). By Theorem 2.1 there exists a path α from γ to y , where y is a reparametrization of x , with L L ( ds ) ( α ) < ε . Now let θ ∈ Diff( S )such that y ( u ) = x ( θ ( u )) and define a path α from x to y by c ( t, u ) := (1 − t ) u + tθ ( u ) α ( t, u ) := x ( c ( t, u ))Then ∂ t α = x (cid:48) ( c ) c t = x (cid:48) ( c )( θ ( u ) − u ) ∂ u α = x (cid:48) ( c ) c u and by (8) the L ( ds )-length of α is L L ( ds ) ( α ) = (cid:90) (cid:18)(cid:90) S (cid:12)(cid:12) x (cid:48) ( c ) (cid:12)(cid:12) | θ ( u ) − u | dc (cid:19) / dt ≤ (cid:107) x (cid:48) (cid:107) / L ∞ ≤ ε α with α ( − t ) to form a path p from γ to x with L L ( ds ) ( p ) < ε .By the same method we construct a path q from γ to x with L L ( ds ) ( q ) < ε and thenthe concatenation of p with q ( − t ) is a path from γ to γ with arbitrarily small L ( ds )-length. We assumed γ , γ were smooth, but since the smooth curves are dense in C and (cid:107) α t (cid:107) L ( ds ) ≤ c (cid:107) α t (cid:107) C (cid:107) α u (cid:107) L for any path α , we can also join any pair of curves in Imm by a path with arbitrarily small L ( ds )-length. (cid:3) Remark . As mentioned in the introduction, an alternative proof of a more generalresult is outlined in [5]. The proof relies on another theorem of Michor and Mumford [20](extended in eg. [8]) showing that the right invariant L metric on diffeomorphism groupsgives vanishing distance.3. Symmetries of metrics and gradient flows
The standard curve shortening flow (3) enjoys several important symmetries: • Isometry of the plane: if A : R → R is an isometry and X : S × [0 , T ) → R isa solution to curve shortening flow then A ◦ X is also a solution. • Reparametrisation: if φ ∈ Diff( S ) and X ( u, t ) is a solution to curve shorteningflow then X ( φ ( u ) , t ) is also a solution. • Scaling spacetime: if X ( u, t ) is a solution to curve shortening flow then so is λX ( u, t/λ ) , with λ > It is interesting to note that these symmetries can be observed directly from symmetriesof the length functional L and the H ( ds ) Riemmannian inner product without actuallycalculating the gradient. Lemma 3.1.
Suppose there is a free group action of G on ( M, g ) which is an isometry ofthe Riemannian metric g and which leaves E : M → R invariant. Then the gradient flowof E with respect to g is invariant under the action. Proof.
Since E ( x ) = E ( λx ) for all λ ∈ G we have dE x = dE λx dλ x then equating dE x V = (cid:104) grad E x , V (cid:105) x = (cid:104) dλ x grad E x , dλ x V (cid:105) λx dE λx dλ x V = (cid:104) grad E λx , dλ x V (cid:105) λx shows that dλ x grad E x = grad E λx ( dλ x has full rank because the action is free). Thereforeif X is a solution to X t = − grad E X then( λX ) t = − dλ X grad E X = − grad E λX so λX is also a solution. (cid:3) To demonstrate we observe the following symmetries of the H ( ds ) gradient flow. Isometry . An isometry A : R → R induces A : Imm ( S , R ) → Imm ( S , R ) by Aγ = A ◦ γ . Since an isometry is length preserving L ( Aγ ) = L ( γ )= ⇒ d L Aγ dA γ = d L γ (9)and similarly for the arc-length functions s Aγ = s γ = ⇒ ds Aγ = ds γ . Hence, in the H ( ds ) metric: (cid:104) dA γ ( ξ ) , dA γ ( η ) (cid:105) Aγ = (cid:90) (cid:104) dA γ ( ξ ) , dA γ ( η ) (cid:105) ds Aγ + (cid:90) (cid:104) dds dA γ ( ξ ) , dds dA γ ( η ) (cid:105) ds Aγ = (cid:104) ξ, η (cid:105) γ i.e. the induced map A : Imm ( S , R ) → Imm ( S , R ) is an H ( ds ) isometry. Now byLemma 3.1 if X is a solution of the H ( ds ) gradient flow of L then so is AX . Reparametrisation.
Given φ ∈ Diff( S ) we have L ( γ ) = L ( γ ◦ φ ) and the map Φ( γ ) = γ ◦ φ is linear on Imm k ( S , R ) so d L γ = d L Φ γ Φ. Assuming for simplicity that φ (cid:48) > (cid:104) Φ ξ, Φ η (cid:105) Φ γ = (cid:90) S (cid:104) ξ ( φ ( u )) , η ( φ ( u )) (cid:105)| γ (cid:48) ( φ ) φ (cid:48) ( u ) | du + (cid:90) S (cid:68) | y (cid:48) ( φ ) φ (cid:48) ( u ) | ddu ξ ( φ ( u )) , | y (cid:48) ( φ ) φ (cid:48) ( u ) | ddu η ( φ ( u )) (cid:69) | γ (cid:48) ( φ ) φ (cid:48) ( u ) | du = (cid:90) S (cid:104) ξ ( φ ) , η ( φ ) (cid:105)| γ (cid:48) ( φ ) | dφ + (cid:90) S (cid:68) | y (cid:48) ( φ ) | ddφ ξ ( φ ) , | y (cid:48) ( φ ) | ddφ η ( φ ) (cid:69) | γ (cid:48) ( φ ) | dφ = (cid:104) ξ, η (cid:105) γ so Φ is also an H ( ds ) isometry and again by Lemma 3.1 the gradient flow is invariantunder reparametrisation. N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 11 Scaling space-time.
For a dilation of R by λ > L ( λx ) = λ L ( x ). To get a space-time scaling symmetry we need the metric to also be homogeneous.However the H ( ds ) metric is not homogeneous: (cid:104) λξ, λη (cid:105) λγ = (cid:90) (cid:104) λξ, λη (cid:105) λds γ + (cid:90) (cid:28) λ dds γ λξ, λ dds γ λη (cid:29) λds γ = λ (cid:90) (cid:104) ξ, η (cid:105) ds + λ (cid:90) (cid:104) ξ s , η s (cid:105) ds If we want space-time scaling we need to use a different metric. For example the metric (cid:104) ξ, η (cid:105) H ( d ¯ s ) := (cid:104) h, k (cid:105) L ( d ¯ s ) + L (cid:104) ξ s , η s (cid:105) L ( d ¯ s ) (10)which is used in [28] satisfies (cid:104) λξ, λη (cid:105) λγ = λ (cid:104) ξ, η (cid:105) γ , and then d L λγ ξ = (cid:104) grad L λγ , ξ (cid:105) λγ = (cid:104) grad L λγ , ξ (cid:105) γ d L γ ξ = (cid:104) grad L γ , ξ (cid:105) γ Since L ( λγ ) = λ L ( γ ) we have d L λγ λv = ∂∂t L ( λγ + tλv ) = λd L γ v i.e. d L λγ = d L γ and so from above grad L λγ = grad L γ . Now if X ( u, t ) is a solution to X t = − grad L X then defining ˜ X ( u, t ) := λX ( u, t/λ ) we have˜ X t ( u, t ) = X t ( u, t/λ ) = − grad L X ( u,t/λ ) = − grad L ˜ X The gradient flow for length with respect to the H ( ds ) Riemannianmetric
In this section our focus is on the H ( ds )-gradient flow for length on a variety of spaces.4.1. Derivation, stationary solutions and circles.
The H ( ds ) gradient of length isdefined by d L γ V = (cid:104) grad L γ , V (cid:105) H ( ds ) = (cid:90) (cid:104) grad L γ , V (cid:105) ds + (cid:90) (cid:104) (grad L γ ) s , V s (cid:105) ds = (cid:90) (cid:104) grad L γ − (grad L γ ) ss , V (cid:105) ds Comparing with (2) the gradient of length with respect to the H ( ds ) metric must satisfyweakly(11) (grad L γ ) ss − grad L γ = dTds . We solve this ODE in arc-length parametrisation using the Green’s function method.Considering(12) G ss ( s, ˜ s ) − G ( s, ˜ s ) = δ ( s − ˜ s )(again weakly) with C -periodic boundary conditions and the required discontinuity wefind the Green’s function(13) G ( s, ˜ s ) = cosh (cid:0) | s − ˜ s | − L (cid:1) − L ) . (cf. [28] eqn. (12) for the metric (10) above.) Then the solution to (11) isgrad L γ ( s ) = (cid:90) L dTd ˜ s G ( s, ˜ s ) d ˜ s . (14) We can integrate by parts twice in (14) to obtaingrad L γ ( s ) = γ ( s ) + (cid:90) L γ (˜ s ) G ( s, ˜ s ) d ˜ s and we observe that it is not neccesary for γ to have a second derivative. Indeed, usingintegration by parts and (12) we find (cid:104) grad L γ , V (cid:105) H ( ds ) = (cid:90) (cid:28) γ + (cid:90) γG d ˜ s, V (cid:29) + (cid:28) γ s + (cid:90) γG s d ˜ s, V s (cid:29) ds = (cid:90) (cid:28) γ + (cid:90) γ ( G − G ss ) d ˜ s, V (cid:29) + (cid:90) (cid:104) γ s , V s (cid:105) ds = (cid:90) (cid:104) γ s , V s (cid:105) ds = d L γ V Definition.
Consider a family of curves X : S × ( a, b ) → R where for each t ∈ ( a, b ) ⊂ R , X ( · , t ) ∈ Imm . We term X an H ( ds ) curve shortening flow if(15) ∂ t X ( s, t ) = − X ( s, t ) − (cid:90) L X (˜ s, t ) G ( X ; s, ˜ s ) d ˜ s where G is given by(16) G ( X ; s, ˜ s ) = cosh (cid:0) | s − ˜ s | − L (cid:1) − L ) for 0 ≤ s, ˜ s ≤ L . Here we write G ( X ; s, ˜ s ) to emphasize the dependence on the curve X ( ., t ), but henceforthwe will omit the first argument unless it is needed to avoid ambiguity. Remark . Note that (15) makes sense on the larger space H ( S , R ) \ C where C := { X ∈ H ( S , R ) : (cid:107) X (cid:48) ( u ) (cid:107) L = 0 } is the space of constant maps, provided we do not use the arc length parametrisation.That is, we consider(17) ∂ t X ( u, t ) = F ( X ( u, t ))where F is defined by F ( x ; u ) := − x ( u ) − (cid:90) x (˜ u ) G ( x ; u, ˜ u ) | x (cid:48) (˜ u ) | d ˜ uG ( x ; u, ˜ u ) := cosh (cid:16) | s x ( u ) − s x (˜ u ) | − L ( x )2 (cid:17) (cid:16) − L ( x )2 (cid:17) (18)The constant maps are problematic: viewing G as a map from H ( S , R ) × S × S → R we see that taking a sequence in the first variable toward the space of constant mapsresults in −∞ . Then in the evolution equation (15), the integral involving G along such asequence is not well-defined. Most of the results that follow will be proved for this largerspace H ( S , R ) \ C . However, the interpretation of this flow as the H ( ds ) gradient flowof length requires that we use the space Imm . This is so that H ( ds ) is a Riemannianmetric: the product (7) is not positive definite at curves which are not immersed, and infact is not necessarily well-defined because of the arc length derivatives. Moreover, L is notdifferentiable outside of Imm . Nevertheless we proceed to study the flow mostly in thespace H ( S , R ) \C , but bear in mind that on this space it is some kind of pseudo-gradient. N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 13 We begin our study of the flow by considering stationary solutions and observing theevolution of circles.
Lemma 4.2.
There are no stationary solutions to the H ( ds ) curve shortening flow. Proof.
From (15), a map X ∈ H ( S , R ) is stationary if X ( s ) = − (cid:90) L X (˜ s ) G ( s, ˜ s ) d ˜ s . The arc-length function is in H ( S , R ) and so G (see (16)) is in turn in H ( S , R ).Differentiating, we find X s ( s ) = − (cid:90) L X (˜ s ) G s ( s, ˜ s ) d ˜ s , and so the first derivative of X exists classically. Iterating this with integration by partsshows that in fact all derivatives of X exist and it is a smooth map.Furthermore, examining the case of the second derivative in detail, we find (applying(12))(19) X ss = − (cid:90) L X (˜ s ) G ss ( s, ˜ s ) d ˜ s = − X ( s ) − (cid:90) L X (˜ s ) G ( s, ˜ s ) d ˜ s = 0 . Since X is periodic, this implies that X must be the constant map. As explained in remark4.1, G is singular at constant maps. (cid:3) Let us now consider the case of a circle. Here, we see a stark difference to the case ofthe classical L ( ds ) curve shortening flow. Lemma 4.3.
Under the H ( ds ) curve shortening flow, an initial circle in H ( S , R ) withany radius and any centre will(i) Exist for all time; and(ii) Shrink homothetically to a point as t → ∞ . Proof.
We can immediately conclude from the symmetry of the length functional and thesymmetry of the circle that the flow must evolve homothetically (see Lemma 3.1). Wemust calculate the evolution of the radius of the circle.So, suppose that X is an H ( ds ) curve shortening flow of the form(20) X ( u, t ) = r ( t )(cos u, sin u )with r (0) >
0. Here u is the arbitrary parameter and not the arclength variable. Then X ( s, t ) = r ( t )(cos( sr ) , sin( sr )) and X ss = − r X . Therefore X t ( s, t ) = − (cid:90) X ˜ s ˜ s (˜ s, t ) G ( s, ˜ s ) d ˜ s = 1 r (cid:90) X (˜ s, t ) G ( s, ˜ s ) d ˜ s . Applying (12) and integrating by parts gives X t = − r X t − r X .
Differentiating (20) gives X t = ˙ rr X and then substituting into the above leads to the ODEfor r ( t ): ˙ r = − rr + 1 . Using separation of variables yields r e r = e − t + c (here c = log( r (0) e r (0) )) which hassolutions r ( t ) = ± (cid:112) W ( e c − t )where W is the Lambert W function (the inverse(s) of xe x ). Since t (cid:55)→ (cid:112) W ( e c − t ) is amonotonically decreasing function converging to zero as t → ∞ , this finishes the proof. (cid:3) Existence and uniqueness.
Now we turn to establishing existence and uniquenessfor the H ( ds ) curve shortening flow.For the initial data, we take it to be on the largest possible space for which (15) makessense . As explained in Remark 4.1, this is the space of maps H ( S , R ) \ C where C := { X ∈ H ( S , R ) : (cid:107) X (cid:48) ( u ) (cid:107) L = 0 } is the space of constant maps. We note that C is generated by the action of translationsin R applied to the orbit of the diffeomorphism group at any particular constant mapin H ( S , R ). Since the orbit of the diffeomorphism group applied to a constant map istrivial, the space C turns out to be two-dimensional only.The main result of this section is the following. Theorem 4.12.
For each X ∈ H ( S , R ) \ C there exists a unique eternal H ( ds ) curveshortening flow X : S × R → R in C ( R ; H ( S , R ) \ C ) such that X ( · ,
0) = X . This is proven in two parts.4.2.1.
Local existence.
We begin with a local existence theorem.
Theorem 4.4.
For each X ∈ H ( S , R ) \ C there exists a T > and unique H ( ds ) curve shortening flow X : S × [ − T , T ] → R in C ([ − T , T ]; H ( S , R ) \ C ) such that X ( · ,
0) = X . The flow (15) is essentially a first-order ODE and so we will be able to establish thisresult by applying the Picard-Lindel¨of theorem in H ( S , R ) \ C . (see [33, Theorem3.A]). Note that this means we should not expect any kind of smoothing effect or otherphenomena associated with diffusion-type equations such as the L ( ds ) curve shorteningflow. Of course, we will need to show that the flow a-priori remains away from theproblematic set C .Recalling (18), we observe the following regularity for F in our setting. Lemma 4.5.
For any x ∈ H ( S , R ) \ C consider F and G as defined in (18). Then F ( x ) ∈ H ( S , R ). Proof.
The weak form of equation (12) implies continuity and symmetry of the Greensfunction G , as well as (cid:90) L G ( s, ˜ s ) ds = − . N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 15 Note that there is a discontinuity in the first derivative of G with respect to either variable.Since G is strictly negative we have (cid:82) L | G ( s, ˜ s ) | d ˜ s = 1. Now F ( x ) = − x − (cid:90) L x (˜ s ) G ( s, ˜ s ) d ˜ s = (cid:90) L ( x ( s ) − x (˜ s )) G ( s, ˜ s ) d ˜ s = (cid:90) L (cid:90) s ˜ s x s ds G ( s, ˜ s ) d ˜ s therefore | F ( x ) | ≤ (cid:90) L | s − ˜ s || G ( s, ˜ s ) | d ˜ s ≤ L ( x )(21)and we have(22) (cid:107) F ( x ) (cid:107) L ( du ) ≤ L ( x ) . For the derivative with respect to u , using G s = − G ˜ s and integration by parts we find F ( x ; u ) u = − x u − | x u | (cid:90) L x (˜ s ) G s ( s, ˜ s ) d ˜ s = − x u − | x u | (cid:90) L x ˜ s (˜ s ) G ( s, ˜ s ) d ˜ s . (23)This implies(24) (cid:107) F ( x ) u (cid:107) L ≤ (cid:107) x u (cid:107) L Since our convention is that | S | = 1, we have L ( x ) ≤ (cid:107) x (cid:107) L , and so the inequalities (22)and (24) together show that F ( x ) ∈ H ( S , R ). (cid:3) The H regularity of F from Lemma 4.5 is locally uniform (for given initial data), witha Lipschitz estimate in H , as the following lemma shows. Lemma 4.6.
Given x ∈ H ( S , R ) \ C let Q b = { x ∈ H ( S , R ) \ C : (cid:107) x − x (cid:107) H ≤ b < L ( x ) } where b > L ≥ K > (cid:107) x (cid:107) H such that (cid:107) F ( x ) (cid:107) H < K, for all x ∈ Q b (cid:107) F ( x ) − F ( y ) (cid:107) H ≤ L (cid:107) x − y (cid:107) H , for all x, y ∈ Q b Proof.
To obtain the estimates and remain away from the problematic set C it is necessarythat the length of each x ∈ Q b is bounded away from zero. This is the reason for theupper bound on b . Indeed if x ∈ Q b then (note that | S | = 1 in our convention) (cid:12)(cid:12) (cid:107) x (cid:48) (cid:107) L − (cid:107) x (cid:48) (cid:107) L (cid:12)(cid:12) ≤ (cid:107) x (cid:48) − x (cid:48) (cid:107) L ≤ (cid:107) x (cid:48) − x (cid:48) (cid:107) L ≤ (cid:107) x − x (cid:107) H ≤ b hence L ( x ) − b ≤ L ( x ) ≤ L ( x ) + b (25)and L ( x ) − b > G ( x ; u, ˜ u ) exists on Q b and since s x ( u ) ≤ L ( x ) we deduce(26) | G ( x ; u, ˜ u ) | ≤ cosh( L ( x )2 )2 sinh( L ( x )2 ) = 12 coth (cid:16) L ( x )2 (cid:17) . We will also need the derivative(27) ∂ u G ( x ; u, ˜ u ) = sinh (cid:16) | s x ( u ) − s x (˜ u ) | − L ( x )2 (cid:17) − L ( x )2 ) sgn( u − ˜ u ) | x (cid:48) ( u ) | which obeys the estimate(28) | ∂ u G ( x ; u, ˜ u ) | ≤ | x (cid:48) ( u ) | . Since G is well-defined on Q b , we may use (22) and (24) from the proof of Lemma 4.5 toobtain(29) (cid:107) F ( x ) (cid:107) H ≤ c (cid:107) x (cid:107) H ≤ cb + c (cid:107) x (cid:107) H =: K for a constant c > x ∈ Q b .As for the Lipschitz estimate, we will begin by studying the Lipschitz property for G .First note that the arc length function is Lipschitz as a function on H ( S , R ): | s x ( u ) − s y ( u ) | ≤ (cid:90) u (cid:12)(cid:12) | x (cid:48) ( a ) | − | y (cid:48) ( a ) | (cid:12)(cid:12) da ≤ (cid:90) u (cid:12)(cid:12) x (cid:48) ( a ) − y (cid:48) ( a ) (cid:12)(cid:12) da ≤ (cid:107) x − y (cid:107) H and setting u = 1 we also have |L ( x ) − L ( y ) | ≤ (cid:107) x − y (cid:107) H . For the numerator of G , note that cosh is smooth and its domain here is bounded via (25),and so there is a c > (cid:12)(cid:12)(cid:12)(cid:12) cosh (cid:18) | s x ( u ) − s x (˜ u ) | − L ( x )2 (cid:19) − cosh (cid:18) | s y ( u ) − s y (˜ u ) | − L ( y )2 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:12)(cid:12)(cid:12)(cid:12) | s x ( u ) − s x (˜ u ) | − L ( x )2 − | s y ( u ) − s y (˜ u ) | + L ( y )2 (cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:107) x − y (cid:107) H A similar argument applies to the denominator sinh( −L ( x ) /
2) and moreover inequality(25) ensures that sinh( −L ( x ) /
2) is bounded away from zero. Since the quotient of twoLipschitz functions is itself Lipschitz provided the denominator is bounded away from zero,we have that G is Lipschitz, i.e. there is a constant c > | G ( x ; u, ˜ u ) − G ( y ; u, ˜ u ) | ≤ c (cid:107) x − y (cid:107) H . Now we have F ( x )( u ) − F ( y )( u )= − x ( u ) + y ( u ) − (cid:90) x (˜ u ) G ( x ; u, ˜ u ) | x (cid:48) (˜ u ) | − y (˜ u ) G ( y ; u, ˜ u ) | y (cid:48) (˜ u ) | d ˜ u = − ( x ( u ) − y ( u )) − (cid:90) ( x − y ) G ( x ) | x (cid:48) | + yG ( x ) (cid:0) | x (cid:48) | − | y (cid:48) | (cid:1) + y | y (cid:48) | ( G ( x ) − G ( y )) d ˜ u . N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 17 By (25) and (26) there exists c such that | G ( x ; u, ˜ u ) | ≤ c for all x ∈ Q b . Then using theLipschitz condition for G we find | F ( x ) − F ( y ) |≤ | x − y | + (cid:90) c | x − y || x (cid:48) | + c | y || x (cid:48) − y (cid:48) | + c | y || y (cid:48) |(cid:107) x − y (cid:107) H d ˜ u ≤ | x − y | + c (cid:107) x − y (cid:107) L (cid:107) x (cid:48) (cid:107) L + c (cid:107) y (cid:107) L (cid:107) x (cid:48) − y (cid:48) (cid:107) L + c (cid:107) y (cid:107) L (cid:107) y (cid:48) (cid:107) L (cid:107) x − y (cid:107) H Therefore, recalling that x, y ∈ Q b satisfy (cid:107) x (cid:107) , (cid:107) y (cid:107) ≤ (cid:107) x (cid:107) + b , integrating the above gives(30) (cid:107) F ( x ) − F ( y ) (cid:107) L ≤ const (cid:107) x − y (cid:107) H We need a similar result for(31) F ( x ) (cid:48) ( u ) − F ( y ) (cid:48) ( u ) = − x (cid:48) ( u ) + y (cid:48) ( u ) − (cid:90) ( x − y ) ∂ u G ( x ) (cid:107) x (cid:48) (cid:107) + y∂ u G ( x ) (cid:0) (cid:107) x (cid:48) (cid:107) − (cid:107) y (cid:48) (cid:107) (cid:1) + y (cid:107) y (cid:48) (cid:107) ( ∂ u G ( x ) − ∂ u G ( y )) d ˜ u Comparing (27), define A ( x ; u, ˜ u ) := sinh (cid:16) | s x ( u ) − s x (˜ u ) | − L ( x )2 (cid:17) − L ( x )2 )so that ∂ u G ( u, ˜ u ) = A ( x ; u, ˜ u ) (cid:107) x (cid:48) ( u ) (cid:107) sgn( u − ˜ u ) . Then as in (28) we have | A ( x )( u, ˜ u ) | ≤ and arguing as for G above we also have that A is Lipschitz: | A ( x ; u, ˜ u ) − A ( y ; u, ˜ u ) | ≤ const (cid:107) x − y (cid:107) H Now (cid:90) | ∂ u G ( x ) − ∂ u G ( y ) | d ˜ u = (cid:90) (cid:12)(cid:12)(cid:12) A ( x ) (cid:12)(cid:12) x (cid:48) ( u ) (cid:12)(cid:12) − A ( y ) (cid:12)(cid:12) y (cid:48) ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜ u = (cid:90) (cid:12)(cid:12)(cid:12) ( A ( x ) − A ( y )) (cid:12)(cid:12) x (cid:48) ( u ) (cid:12)(cid:12) + A ( y ) (cid:0)(cid:12)(cid:12) x (cid:48) ( u ) (cid:12)(cid:12) − (cid:12)(cid:12) y (cid:48) ( u ) (cid:12)(cid:12)(cid:1)(cid:12)(cid:12)(cid:12) d ˜ u ≤ (cid:12)(cid:12) x (cid:48) ( u ) (cid:12)(cid:12) (cid:90) | A ( x ) − A ( y ) | d ˜ u + (cid:12)(cid:12) x (cid:48) ( u ) − y (cid:48) ( u ) (cid:12)(cid:12) (cid:90) | A ( y ) | d ˜ u ≤ const (cid:12)(cid:12) x (cid:48) ( u ) (cid:12)(cid:12) (cid:107) x − y (cid:107) H + (cid:12)(cid:12) x (cid:48) ( u ) − y (cid:48) ( u ) (cid:12)(cid:12) . Using this estimate in (31), together with (28) gives (cid:12)(cid:12) F ( x ) (cid:48) ( u ) − F ( y ) (cid:48) ( u ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) x (cid:48) ( u ) − y (cid:48) ( u ) (cid:12)(cid:12) + (cid:12)(cid:12) x (cid:48) ( u ) (cid:12)(cid:12) (cid:107) x − y (cid:107) L (cid:107) x (cid:48) (cid:107) L + (cid:12)(cid:12) x (cid:48) ( u ) (cid:12)(cid:12) (cid:107) y (cid:107) L (cid:107) x (cid:48) − y (cid:48) (cid:107) L + (cid:0) const (cid:12)(cid:12) x (cid:48) ( u ) (cid:12)(cid:12) (cid:107) x − y (cid:107) H + (cid:12)(cid:12) x (cid:48) ( u ) − y (cid:48) ( u ) (cid:12)(cid:12)(cid:1) (cid:107) y (cid:107) L (cid:107) y (cid:48) (cid:107) L and then(32) (cid:107) F ( x ) (cid:48) − F ( y ) (cid:48) (cid:107) L ≤ (cid:107) x − y (cid:107) L + (cid:107) x (cid:48) (cid:107) L (cid:107) x − y (cid:107) L + (cid:107) x (cid:48) (cid:107) L (cid:107) y (cid:107) L (cid:107) x (cid:48) − y (cid:48) (cid:107) L + const (cid:107) x (cid:48) (cid:107) L (cid:107) y (cid:107) L (cid:107) y (cid:48) (cid:107) L (cid:107) x − y (cid:107) H + (cid:107) x (cid:48) − y (cid:48) (cid:107) L (cid:107) y (cid:107) L (cid:107) y (cid:48) (cid:107) L ≤ const (cid:107) x − y (cid:107) H . Combining (30) and (32) gives the required estimate, there exists L such that (cid:107) F ( x ) − F ( y ) (cid:107) H ≤ L (cid:107) x − y (cid:107) H for all x, y ∈ Q b . (cid:3) Proof of Theorem 4.4.
According to the generalised (to Banach space) Picard-Lindel¨oftheorem in [33] (Theorem 3.A), the estimates in Lemma 4.6 guarantee existence anduniqueness of a solution on the interval provided KT < b . (cid:3) Global existence.
We may extend the existence interval by repeated applications ofthe Picard-Lindel¨of theorem from [33].There are two issues to be resolved for this. First, the constants K and L from Lemma4.6 depend on the H -norm of x . When we attempt to continue the solution, we mustshow that in the forward and backward time directions this norm does not explode infinite time to + ∞ .Second, the flow must remain within Q b for some b ; as the evolution continues forward,length is decreasing, and so the amount of time that we can extend depends not only onthe H -norm of the solution but also the length bound from below. In the backward timedirection length is in fact increasing , so this second issue does not arise there.First, we study the L ∞ -norm of the solution. Lemma 4.7.
Let X be an H ( ds ) curve shortening flow defined on some interval ( − T, T ).Then (cid:107) X ( t ) (cid:107) ∞ is non-increasing on ( − T, T ). Furthermore, we have the estimate (cid:107) X ( t ) (cid:107) ∞ ≤ e − t (cid:107) X (0) (cid:107) ∞ , for t < . Proof.
In the forward time direction, we proceed as follows for the uniform bound. Forany t ∈ ( − T, T ) there exists u such that (cid:107) X ( t ) (cid:107) ∞ = | X ( t , u ) | and then ddt | X ( t, u ) | (cid:12)(cid:12)(cid:12)(cid:12) ( t ,u ) = 2 (cid:104) X ( t , u ) , X t ( t , u ) (cid:105) = − | X ( t , u ) | − (cid:28) X ( t , u ) , (cid:90) X ( t , ˜ s ) G ( s , ˜ s ) d ˜ s (cid:29) = − (cid:107) X ( t ) (cid:107) ∞ − (cid:28) X ( t , u ) , (cid:90) X ( t , ˜ s ) G ( s , ˜ s ) d ˜ s (cid:29) ≤ − (cid:107) X ( t ) (cid:107) ∞ + 2 (cid:107) X ( t ) (cid:107) ∞ (cid:90) G ( s , ˜ s ) d ˜ s ≤ . Now let t = sup { t ≥ t : (cid:107) X ( t ) (cid:107) L ∞ = | X ( t, u ) |} . By the inequality above (cid:107) X ( t ) (cid:107) ∞ isnon-increasing for all t ∈ [ t , t ), and by the continuity of X in t , lim t → t (cid:107) X ( t ) (cid:107) L ∞ = (cid:107) X ( t ) (cid:107) L ∞ . Since t was arbitrary, it follows that (cid:107) X ( t ) (cid:107) ∞ cannot increase at any t .In the backward time direction, we need an estimate from below. Let us calculate ddt (cid:0) e t X ( t, u ) (cid:1) = e t (cid:16) ( − X ( t, u ) + X ( t, u )) − (cid:90) X ( t, ˜ s ) G ( s, ˜ s ) d ˜ s (cid:17) = − e t (cid:90) X ( t, ˜ s ) G ( s, ˜ s ) d ˜ s N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 19 so ( u as before) ddt (cid:0) e t | X ( t, u ) | (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ( t ,u ) = − e t (cid:28) X ( t , u ) , (cid:90) X ( t , ˜ s ) G ( s , ˜ s ) d ˜ s (cid:29) ≥ − e t (cid:107) X ( t ) (cid:107) ∞ = − e t | X ( t , u ) | . Hence ddt (cid:0) e t | X ( t, u ) | (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) ( t ,u ) ≥ t to 0 (assuming t <
0, and u changing as necessary) this translatesto (cid:107) X ( t ) (cid:107) ∞ ≤ e − t (cid:107) X (0) (cid:107) ∞ This is the claimed estimate in the statement of the lemma. (cid:3)
Lemma 4.8.
Let X be an H ( ds ) curve shortening flow defined on some interval ( − T, T ).Then (cid:107) X u ( t ) (cid:107) L is non-increasing on ( − T, T ). Furthermore, we have the estimate (cid:107) X u ( t ) (cid:107) L ≤ e − t (cid:107) X u (0) (cid:107) L , for t < . Proof.
As in (23) X tu = − X u − | X u ( u ) | (cid:90) L X ˜ s G d ˜ s and therefore, recalling that (cid:82) L | G ( s, ˜ s ) | d ˜ s = 1, ddt (cid:90) | X u | du = 2 (cid:90) (cid:104) X ut , X u (cid:105) du = − (cid:90) (cid:104) X u , X u (cid:105) du − (cid:90) (cid:28) X u , | X u | (cid:90) L X ˜ s G d ˜ s (cid:29) du ≤ − (cid:107) X u (cid:107) L + 2 (cid:90) | X u | (cid:90) L | G | d ˜ s du ≤ . This settles the forward time estimate. As before, for the backward time estimate we needa lower bound. We calculate ddt (cid:18) e t (cid:90) | X u | du (cid:19) = 2 e t (cid:90) (cid:28) X u , | X u | (cid:90) L X ˜ s G d ˜ s (cid:29) du ≥ − e t (cid:107) X u (cid:107) L . The same integration as in the backward time estimate for Lemma 4.7 yields the claimedbackward in time estimate. (cid:3)
The estimates of Lemmata 4.7, 4.8 yield the following control on the H -norm of thesolution. Corollary 4.9.
Let X be an H ( ds ) curve shortening flow defined on some interval( − T, T ). Then (cid:107) X ( t ) (cid:107) H ≤ (cid:107) X (0) (cid:107) H , for all t ∈ [0 , T ) , and (cid:107) X ( t ) (cid:107) H ≤ (cid:107) X (0) (cid:107) H e − t , for all t ∈ ( − T, . A similar technique allows us to show also that if the initial data for the flow is animmersion, it remains an immersion.
Lemma 4.10.
Let X be an H ( ds ) curve shortening flow defined on some interval ( − T, T )with X (0) ∈ Imm . Then X ( t ) ∈ Imm for all t ∈ ( − T, T ). Proof.
Using (23) we have(33) ddt | X u | = 2 (cid:104) X ut , X u (cid:105) = − | X u | − | X u |(cid:104) X u , (cid:90) L X ˜ s G d ˜ s (cid:105) . Now since (cid:82) L X s ds = 0 we have (cid:90) L X s (˜ s ) G ds = (cid:90) L X s (˜ s ) G ( s, ˜ s ) d ˜ s − G ( s, s ) (cid:90) L X s (˜ s ) d ˜ s = (cid:90) L X s ( G ( s, ˜ s ) − G ( s, s )) d ˜ s = (cid:90) L X s (cid:90) ˜ ss G τ ( s, τ ) dτ d ˜ s so using | G s | ≤ (cf. (28)) we find(34) (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) L X s (˜ s ) G ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) | s − ˜ s | d ˜ s ≤ L / . Using this estimate with (33) yields( − − L ) | X u | ≤ ddt | X u | ≤ ( − L ) | X u | . (35)From d L dt = −(cid:107) grad L X (cid:107) H ( ds ) we know that L is non-increasing and so rearranging theinequality on the left and multiplying by an exponential factor gives0 ≤ ddt (cid:16) e (2+ L (0) ) t | X u | (cid:17) . Now integrating from 0 to t > | X u (0 , u ) | e − (2+ L (0) ) t ≤ | X u ( t, u ) | Then since X u is initially an immersion, it remains so for t >
0. The estimate backwardin time is analogous but we instead use the second inequality in (35) and integrate from t < | X u (0 , u ) | e − ( L (0) − t ≤ | X u ( t, u ) | for t < (cid:3) Corollary 4.11.
Let X be an H ( ds ) curve shortening flow defined on some interval( − T, T ) with
T < ∞ . Then there exists ε > L ( X ( t )) > ε for all t ∈ ( − T, T ). Proof.
For t ≥
0, taking the square root in (36) and then integrating over u gives L (0) e − (1+ L (0) / t ≤ L ( t ) . Since L ( t ) is non-increasing the result follows. (cid:3) Theorem 4.12.
Any H ( ds ) curve shortening flow defined on some interval ( − T, T ) with T < ∞ may be extended to all t ∈ ( −∞ , ∞ ) . N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 21 Proof.
According to Lemma 4.6 and Theorem 4.4, given x ∈ H ( S , R ), for 0 < ε < L ( x ) K there is a unique solution X ( t, u ) for t ∈ [ t − ε , t + ε ]. Following the standardcontinuation procedure, one takes x ± := X ( t ± ε ) as the initial condition for a newapplication of Theorem 4.4 with existence time ε < L ( x ) /K , and so on.Take T to be the maximal time such that the flow X can be extended forward: t ∈ ( − T, T ). If
T < ∞ , then one or more of the following have occurred: • L ( X ( t )) (cid:38) t (cid:37) T ; • (cid:107) X ( t ) (cid:107) H → ∞ as t (cid:37) T .The first possibility is excluded by Corollary 4.11, and the second is excluded by Corollary4.9 (we use T < ∞ here). This is a contradiction, so we must have that T = ∞ .The argument in the backward time direction is completely analogous: suppose T is themaximal time such that the flow X can be extended backward: t ∈ ( T , T ). If
T > −∞ ,then one or more of the following have occurred: • L ( X ( t )) → t (cid:38) T ; • (cid:107) X ( t ) (cid:107) H → ∞ as t (cid:38) T .The first possibility is excluded by the fact that the flow decreases length. The second isexcluded by Corollary 4.9 (we again use T > −∞ here). This is a contradiction, so wemust have that T = −∞ . (cid:3) Convergence.
In this subsection we examine the forward in time limit for the flow.The backward limit is not expected to have nice properties. One way to see this is in the H ( ds ) length of the tail of an H ( ds ) curve shortening flow. (We will see in Lemma 4.17that the H ( ds ) length of any forward trajectory is finite.) For instance, a circle evolvingunder the flow has radius r ( t ) = (cid:112) W ( e c − t ). The H ( ds ) length is larger than the L ( ds )length, and this grows (as t → −∞ ) linear in W ( e c − t ). This is not bounded, and so inparticular the H ( ds ) length of any negative tail is unbounded.Throughout we let X : ( −∞ , ∞ ) → H ( S , R ) be a solution to the H ( ds ) curveshortening flow (15). We will prove lim t →∞ X ( t ) exists and is equal to a constant map.In order to use the H ( ds ) gradient (see Remark 4.1) we present the proof for the casewhere X ( t ) is immersed, but the results can be extended to the flow in H ( S , R ) \ C asdescribed in Remark 4.20 below.It will be convenient to define K by K ( x ) := cosh( | x | − L / L / , x ∈ [0 , L ] , so that G = − K ( | s − ˜ s | ), K ( x ) > (cid:82) L K ( x ) dx = 1. We take the periodic extensionof K to all of R , which we still denote by K , and then(38) (cid:90) L K ( s − ˜ s ) ds = (cid:90) L− ˜ s − ˜ s K ( x ) dx = 1Define, for any γ ∈ H ( S , R ),( γ ∗ K )( s ) := (cid:90) S γ (˜ s ) K ( s − ˜ s ) d ˜ s . This is a so-called ‘nonlinear’ (in γ ) convolution. We have the following version of Young’sconvolution inequality. Lemma 4.13.
For any γ ∈ H ( S , R ) and K , ∗ as above, we have (cid:107) γ ∗ K (cid:107) L ( ds ) ≤ (cid:107) γ (cid:107) L ( ds ) . (39) Proof.
Write | γ (˜ s ) || K ( s − ˜ s ) | = (cid:16) | γ (˜ s ) | | K ( s − ˜ s ) | (cid:17) | K ( s − ˜ s ) | then by the H¨older inequality and (38) (cid:90) L | γ (˜ s ) || K ( s − ˜ s ) | d ˜ s ≤ (cid:18)(cid:90) | γ (˜ s ) | | K ( s − ˜ s ) | d ˜ s (cid:19) . Hence (cid:107) γ ∗ K (cid:107) L ( ds ) ≤ (cid:90) (cid:18)(cid:90) | γ (˜ s ) || K ( s − ˜ s ) | d ˜ s (cid:19) ds ≤ (cid:90) (cid:90) | γ (˜ s ) | | K ( s − ˜ s ) | d ˜ sds ≤ (cid:90) | γ (˜ s ) | (cid:90) | K ( s − ˜ s ) | ds d ˜ s ≤ (cid:107) γ (cid:107) L ( ds ) . (cid:3) The convolution inequality implies the following a-priori estimate in L . Lemma 4.14.
Let X be an H ( ds ) curve shortening flow. Then (cid:107) X ( t ) (cid:107) L ( ds ) is non-increasing as a function of t . Proof.
First note that since G s = − G ˜ s we have X ts = − X s − (cid:90) L XG s d ˜ s = − X s − (cid:90) L X s (˜ s ) G ds .
Then (using ddt ds = (cid:104) X ts , X s (cid:105) ds ) we find ddt (cid:107) X ( t ) (cid:107) L ( ds ) = 2 (cid:90) L (cid:104) X t , X (cid:105) ds + (cid:90) L | X | (cid:104) X ts , X s (cid:105) ds = − (cid:90) L | X | ds − (cid:90) L (cid:28) X ( s ) , (cid:90) L X (˜ s ) G d ˜ s (cid:29) ds − (cid:90) L | X | ds − (cid:90) L | X | (cid:28) X s , (cid:90) L X s (˜ s ) G d ˜ s (cid:29) ds . H¨older’s inequality and the convolution inequality (39) now yield ddt (cid:107) X ( t ) (cid:107) L ( ds ) ≤ − (cid:90) L | X | ds + 2 (cid:90) L | X || X ∗ K | ds + (cid:90) | X | (cid:90) L | G | d ˜ s ds ≤ . (cid:3) Now we give a fundamental estimate for the H ( ds )-gradient of length along the flow. Lemma 4.15.
Let X be an H ( ds ) curve shortening flow. There exists a constant C > X (0) such that(40) (cid:107) grad H ( ds ) L X ( t ) (cid:107) H ( ds ) ≥ C L ( X ( t )) for all t ∈ [0 , ∞ ). N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 23 Proof.
From (2) and (14), if X is a solution of (15) then ddt L ( X ) = d L X ( − grad H ( ds ) L X )= (cid:90) L (cid:104) X ss ( t, s ) , X ( t, s ) + (cid:90) L X ( t, ˜ s ) G ( s, ˜ s ) d ˜ s (cid:105) ds . Integration by parts with (12) gives ddt L ( X ) = − (cid:90) L (cid:104) X s , X s (cid:105) ds + (cid:90) L (cid:104) X, X (cid:105) ds + (cid:90) L (cid:90) L (cid:104) X ( t, s ) , X ( t, ˜ s ) G ( s, ˜ s ) (cid:105) d ˜ sds = −L ( X ) − (cid:90) L (cid:104) X, X t (cid:105) ds . (41)Since d L dt = −(cid:107) grad H ( ds ) L X (cid:107) H ( ds ) , we have (cid:107) grad H ( ds ) L X (cid:107) H ( ds ) = L − (cid:90) L (cid:68) X, grad H ( ds ) L X (cid:69) ds ≥ L − (cid:107) X (cid:107) L ( ds ) (cid:107) grad H ( ds ) L X (cid:107) L ( ds ) . The inequality 2 ab ≤ εa + ε b for all ε > L ≤ ε (cid:107) X (cid:107) L ( ds ) + (cid:18) ε + 1 (cid:19) (cid:107) grad H ( ds ) L X (cid:107) H ( ds ) ≤ ε (cid:107) X (cid:107) ∞ L + (cid:18) ε + 1 (cid:19) (cid:107) grad H ( ds ) L X (cid:107) H ( ds ) . Now Lemma 4.7 yields L (cid:16) − ε (cid:107) X (0) (cid:107) ∞ (cid:17) ≤ (cid:18) ε + 1 (cid:19) (cid:107) grad H ( ds ) L X (cid:107) H ( ds ) and choosing ε sufficiently small gives (40). (cid:3) The gradient inequality immediately implies exponential decay of length.
Lemma 4.16.
Let X be an H ( ds ) curve shortening flow. The length L ( X ( t )) convergesto zero exponentially fast as t → ∞ . Proof.
Using the gradient inequality (40) we have − d L dt = (cid:107) grad H ( ds ) L X (cid:107) H ( ds ) ≥ C L . Integrating gives(42) L ( t ) ≤ L (0) e − Ct as required. (cid:3) Another consequence of the gradient inequality is boundedness of the H ( ds ) length ofthe positive trajectory X . Lemma 4.17.
Let X be an H ( ds ) curve shortening flow. The H ( ds )-length of { X ( · , t ) : t ∈ (0 , ∞ ) } ⊂ H ( S , R ) is finite. Proof.
From the gradient inequality (40) ddt L = −(cid:107) grad H ( ds ) L X (cid:107) H ( ds ) ≤ −(cid:107) grad H ( ds ) L X (cid:107) H ( ds ) (cid:107) X t (cid:107) H ( ds ) ≤ − C L (cid:107) X t (cid:107) H ( ds ) i.e. ddt (2 L ) ≤ − C (cid:107) X t (cid:107) H ( ds ) and therefore 2 L ( t ) − L (0) ≤ − C (cid:90) t (cid:107) X t (cid:107) H ( ds ) dt , or (cid:90) t (cid:107) X t (cid:107) H ( ds ) dt ≤ L (0) C (43)Taking the limit t → ∞ in the above inequality, the left hand side is the length of thetrajectory X : [0 , ∞ ) → H ( S , R ) measured in the H ( ds ) metric. (cid:3) Now we conclude convergence to a point.
Theorem 4.18.
Let X be an H ( ds ) curve shortening flow. Then X converges as t → ∞ in H to a constant map X ∞ ∈ H ( S , R ) .Proof. Recalling (21) we have | X t | ≤ L and therefore (cid:107) X t (cid:107) H = (cid:90) | X t | du + (cid:90) | X tu | du ≤ L + (cid:90) | X tu | du . (44)Using G s = − G ˜ s we have X ts = − X s − (cid:90) L XG s d ˜ s = − X s − (cid:90) L X s (˜ s ) G ds and then from (34) | X ts | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) L X s (˜ s ) G ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ L ( X ) . (45)Hence | X tu | ≤ | X u | (1 + L / − − L (0) e − Ct ) | X u | ≤ ddt | X u | ≤ ( − L (0) e − Ct ) | X u | (46)Using the second inequality, multiply by the integrating factor e p ( t ) where p ( t ) := (cid:90) t − L (0) e − Cτ dτ, and integrate with respect to t to find | X u ( t ) | ≤ | X u (0) | e − p ( t ) ≤ | X u (0) | e − t + c for some constant c . For future reference we note that the same procedure can be appliedto the lower bound in (46) and then(47) | X u (0) | e − t − c ≤ | X u ( t ) | ≤ | X u (0) | e − t + c . We therefore have | X tu | ≤ | X u (0) | e − t + c (1 + L / N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 25 and then referring back to (44): (cid:107) X t (cid:107) H ≤ L + (cid:107) X u (0) (cid:107) L e − t + c (1 + L / . Using the gradient inequality (40) and monotonicity of L we obtain (cid:107) X t (cid:107) H ≤ c (cid:107) X t (cid:107) H ( ds ) + c e − t + c . By integrating d L dt = −(cid:107) X t (cid:107) H ( ds ) with respect to t we have (cid:90) ∞ (cid:107) X t (cid:107) H ( ds ) dt ≤ L (0) . Hence for all ε > t ε such that (cid:82) ∞ t (cid:107) X t (cid:107) H dt < ε for all t ≥ t ε , and since (cid:13)(cid:13)(cid:13)(cid:13) X ( t ) − X ( t ) (cid:13)(cid:13)(cid:13)(cid:13) H = (cid:13)(cid:13)(cid:13)(cid:13) (cid:90) t t X t dt (cid:13)(cid:13)(cid:13)(cid:13) H ≤ (cid:90) t t (cid:107) X t (cid:107) H dt it follows that X t converges in H to some X ∞ . By (42) the length of X ∞ is zero, i.e. itis a constant map. (cid:3) Remark . If (Imm , H ( ds )) were a complete metric space then Lemma 4.17 would beenough to conclude convergence of the flow. However it is shown in [22] section 6.1 thatthe H ( ds ) geodesic of concentric circles can shrink to a point in finite time, so the spaceis not even geodesically complete. Indeed, Theorem 4.18 demonstrates convergence of theflow with finite path length to a point outside Imm , proving again that (Imm , H ( ds ))is not metrically complete. Remark . For extending the convergence result above to the case of initial data X (0) ∈ H ( S , R ) \ C the main difficulty is that the function F ( X ( u, t )) in (17) is no longerthe H ( ds ) gradient (cf. Remark 4.1) and so we need some other way of arriving at,for example, equation (41). To do this we can approximate by C immersions, as itfollows from eg. Theorem 2.12 in [15] that these are dense in H ( S , R ). Given X ( t ) ∈ H ( S , R ) \ C we let X ε ( t ) be an immersion such that (cid:107) X ( t ) − X ε ( t ) (cid:107) H ≤ ε for t in aneighbourhood of t . Then following (41) we have d L ( X ) dt := lim ε → (cid:18) −L ( X ε ) − (cid:90) L (cid:104) X ε , ( X ε ) t (cid:105) ds (cid:19) . The limit exists because all the terms are bounded by (cid:107) X ε (cid:107) H (for ( X ε ) t this followsfrom (29)). Similarly d L ( X ) dt = −(cid:107) F ( X ) (cid:107) H and we proceed with the rest of the proofs bywriting F ( X ) or X t in place of − grad H ( ds ) L X .5. Shape evolution and asymptotics
Generic qualitative behaviour of the flow.
Computational experiments indicatethat the flow tends to reshape the initial data, gradually rounding out corners and improv-ing regularity. However the scale dependence of the flow introduces an interesting effect:when the length becomes small, the ‘reshaping power’ seems to run out and curves shrinkapproximately self-similarly, preserving regions of low regularity. This means that cornersof small polygons persist whereas corners of large polygons round off under the flow (cf.Figure 1).Heuristically, this is because of the behaviour of G as L →
0. If we Taylor expand G ( s, ˜ s ) ≈ −
12 sinh( L / (cid:18) | s − ˜ s | − L / + . . . (cid:19) since | s − ˜ s | ≤ L , the constant term dominates when L is small. Then X t ≈ − X + (cid:90) L X L / ds = − X + L L /
2) ¯ X and lim L→ L L / = 1 so each point on the curve moves toward its centre.5.2. Remarks on the numerical simulations.
The numerical simulations were carriedout in Julia using a basic forward Euler method. Curves are approximated by polygons.For initial data we take an ordered list of vertices X i in R of a polygon and the lengthof X is of course just the perimeter of the polygon. The arc length s i at X i is the sum ofdistances between vertices up to X i and for the arc-length element ds i we use the averageof the distance to the previous vertex and the distance to the next vertex. The Green’sfunction is then calculated at each pair of vertices: G ij ( X ) = − cosh( | s i − s j | − L ( X ) / L ( X ) / V i at X i is V i = − X i − (cid:88) j X j G ij ( X ) ds j The new position ˜ X i of the vertex X i is calculated by forward-Euler with timestep h :˜ X i = X i + hV i . No efforts were made to quantify errors or test accuracy, but the resultsappear reasonable and stable provided time steps are not too large and there are sufficientlymany vertices. A Jupyter notebook containing the code is available online [26].5.3. Evolution and convergence of an exponential rescaling.Definition.
Let X : [0 , ∞ ) → H ( S , R ) \ C be a solution to the H ( ds ) curve shorteningflow (15). We define the asymptotic profile Y of X as Y ( t, u ) := e t ( X ( t, u ) − X ( t, . We anchor the asymptotic profile so that Y ( t,
0) = 0 for all t . This is not only for con-venience; if the final point that the flow converges to is not the origin, then an unanchoredprofile ˜ Y = e t X would simply disappear at infinity and not converge to anything.The aim in this section is to prove that the asymptotic profile converges. Simulationsindicate that there are a variety of possible shapes for the limit (once we know it exists);numerically, even a simple rescaling of the given initial data may alter the asymptoticprofile. As in the previous section we present the results under the assumption that X is a flow of immersed curves, but they can be extended to H ( S , R ) \ C by the methoddescribed in Remark 4.20.We will need the following refinement of the gradient inequality. Lemma 5.1.
Let X be an H ( ds ) curve shortening flow. For any α ∈ (0 ,
1) there exists t α such that (cid:107) grad H ( ds ) L X (cid:107) H ( ds ) ≥ α L ( X )for all t ≥ t α . Proof.
We abbreviate the gradient to grad L X in order to lighten the notation. Equation(21) implies(48) (cid:107) grad L X (cid:107) L ( ds ) ≤ L ( X )
2N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 27 and therefore from (41) and d L dt = −(cid:107) grad L ( X ) (cid:107) H ( ds ) : (cid:107) grad L X (cid:107) H ( ds ) = L − (cid:90) L (cid:104) X, grad L X (cid:105) ds ≥ L − (cid:107) X (cid:107) ∞ (cid:107) grad L X (cid:107) L ( ds ) ≥ L − (cid:107) X (0) (cid:107) ∞ L where we have also used Lemma 4.7. Now using (42) (cid:107) grad L X (cid:107) H ( ds ) ≥ (1 − (cid:107) X (0) (cid:107) ∞ L (0) e − Ct ) L ( t ) . If α ≥ −(cid:107) X (0) (cid:107) ∞ L (0) we can find the required t α by solving α = 1 −(cid:107) X (0) (cid:107) ∞ L (0) e − Ct α ,otherwise t α = 0. (cid:3) We also need an upper bound for the gradient in terms of length.
Lemma 5.2.
For X ∈ H ( S , R ),(49) (cid:107) grad L X (cid:107) H ( ds ) ≤ L ( X ) + 2 L ( X ) + L ( X ) . Proof.
From (21) we have (cid:107) grad L X (cid:107) L ( ds ) ≤ L . Then (45) implies (cid:107) (grad L X ) s (cid:107) L ( ds ) ≤ (cid:90) (cid:18) L (cid:19) ds ≤ L + L + L (cid:3) We now prove convergence of the asymptotic profile along a subsequence of times –sometimes this is called subconvergence . Theorem 5.3.
Let X be an H ( ds ) curve shortening flow and Y its asymptotic pro-file. There is a non-trivial Y ∞ ∈ C ( S , R ) such that Y ( t ) has a convergent subsequence Y ( t i ) → Y ∞ in C as i → ∞ .Proof. We will show that Y ( t ) is eventually uniformly bounded in H . First we claim thatthere exist constants c , c > t < ∞ such that(50) c < L ( Y ( t )) < c for all t > t . For the upper bound, from L ( Y ) = e t L ( X ), (41) and (48) ddt L ( Y ) = e t L ( X ) + e t ddt L ( X ) = e t (cid:90) L ( X )0 (cid:104) X, grad L X (cid:105) ds ≤ (cid:107) X ( t ) (cid:107) ∞ e t L ( X ) ≤ (cid:107) X (0) (cid:107) ∞ e t L ( X ) (51)From Lemma 5.1, for any α ∈ (0 ,
1) there exists t α such that ddt L ( X ) ≤ − α L ( X ) , t ≥ t α hence L ( X ( t )) ≤ L ( X ( t α )) e − αt for t > t α . Using this in (51) with eg. α = , ddt L ( Y ) ≤ ce − t , t ≥ t / . where c is a constant depending on X (0) and L ( X ( t / )). Integrating from t / to t gives(52) L ( Y ( t )) ≤ L ( Y ( t / )) + 2 ce − t / / − ce − t/
28 P. SCHRADER, G. WHEELER, AND V.-M. WHEELER which gives an upper bound for L ( Y ( t )) for t ≥ t / . For the lower bound the estimate(49) gives − ddt L ( X ) ≤ L ( X ) + 2 L ( X ) + L ( X ) . Let t β be such that L ( X ( t )) < t > t β . (From (42) we can find t β by solving1 = L (0) e − Ct β .)Then also using the gradient inequality (40) there is a constant c such that ddt L ( X ) ≥ −L ( X ) − c L ( X ) (cid:107) grad L X (cid:107) H ( ds ) t > t β . Recalling (49), this implies ( t > max { t β , t / } ) ddt ( e t L ( X )) ≥ − ce t L ( X ) (cid:107) grad L X (cid:107) H ( ds ) ≥ − ˆ ce t L ( X ) ≥ ˜ ce − t . Integrating with respect to t , there is a constant c such that L ( X ) ≥ c e − t , t > max { t β , t / } and therefore L ( Y ) ≥ c for all t > t β . Choosing t to be the greater of t / , t β , we haveestablished the claim (50). We claim also that(53) (cid:107) Y (cid:107) L ≤ c , t > t . To see this, note that by the Fundamental Theorem of Calculus followed by the H¨olderinequality applied to each component of Y : | Y | ≤ L (cid:90) L ( Y )0 | Y s | ds Y = L ( Y )and so (50) gives (cid:107) Y (cid:107) L ≤ c .Multiplying (47) by e t gives(54) | X u (0 , u ) | e − c ≤ | Y u ( t, u ) | ≤ | X u (0 , u ) | e c . We therefore have a uniform bound on (cid:107) Y u (cid:107) L p for 1 ≤ p ≤ ∞ in terms of (cid:107) X u (0) (cid:107) L p . Inparticular, if X (0) ∈ H we have a uniform H bound for Y and then by the Arzela-Ascolitheorem there is a sequence ( t i ) and a Y ∞ ∈ W , ∞ such that Y ( t i ) → Y ∞ in C (cf. [18]Theorems 7.28, 5.37 and the proof of 5.38). (cid:3) This result can be quickly upgraded to full convergence using a powerful decay estimate.
Theorem 5.4.
Let X be an H ( ds ) curve shortening flow and Y its asymptotic profile.There is a non-trivial Y ∞ ∈ H ( S , R ) such that Y ( t ) → Y ∞ in C as t → ∞ .Proof. For the evolution of Y we calculate Y t ( t, u ) = (cid:90) Y ( t, ˜ u )( G ( X ; 0 , s X (˜ u )) − G ( X ; s X ( u ) , s X (˜ u ))) | X ˜ u | d ˜ u . The -Lipschitz property for G (from (28)) implies that (cid:12)(cid:12) ( G ( X ; 0 , s X (˜ u )) − G ( X ; s X ( u ) , s X (˜ u ))) (cid:12)(cid:12) ≤ | s X ( u ) | ≤ L ( Y ( t )) e − t ≤ ce − t , by the estimate (52) in Theorem 5.3. The estimates in the proof of Theorem 5.3 include || Y || ∞ ≤ c . Using these we find | Y t ( t, u ) | = (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) Y ( t, ˜ u )( G ( X ; 0 , s X (˜ u )) − G ( X ; s X ( u ) , s X (˜ u ))) | X ˜ u | d ˜ u (cid:12)(cid:12)(cid:12)(cid:12) ≤ ce − t || Y || ∞ L ( Y ( t )) ≤ ce − t . N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 29 Exponential decay of the velocity implies full convergence by a standard argument (astraightforward modification to C of the C ∞ argument in [3, Appendix A] for instance). (cid:3) The convergence result (Theorem 5.4) applies in great generality. If the initial data X is better than a generic map in H ( S , R ) \ C , for instance if it is an immersion, haswell-defined curvature, or further regularity, then this is preserved by the flow. That claimis proved in the next section (see Theorem 5.6). In these cases, we expect the asymptoticprofile also enjoys these additional properties. This is established in the C space in thesection following that (see Theorem 5.7). Remark . The asymptotic shape is very difficult to determine, in particular, it is notclear if there is a closed-form equation that it must satisfy. As mentioned earlier, we seethis in the numerics. We can also see this in the decay of the flow velocity Y t . It decaysnot because the shape has been optimised to a certain point, but simply because sufficienttime has passed so that the exponential decay terms take over. The asymptotic profile ofthe flow is effectively constrained to a tubular neighbourhood of Y (0).5.4. The H ( ds ) -flow in Imm k spaces. Observe from (21) and (23) that if γ ∈ C thengrad L γ is also C . We might therefore consider the flow with Imm initial data as an ODEon Imm (instead of H ( S , R ) \ C ). In fact the same is true for Imm ( and moreoverImm k ) as we now demonstrate.Assume x ∈ C is an immersion, then(55) G uu = ∂ u ( | x u | G s ) = | x u | G ss − (cid:104) x uu , x s (cid:105) G s and using G ss = G ˜ s ˜ s as well as integrating by parts we obtain(grad L x ) uu = x uu + (cid:90) xG uu d ˜ s = x uu + | x u | (cid:90) x ˜ s ˜ s G d ˜ s − (cid:104) x uu , x s (cid:105) (cid:90) x ˜ s G d ˜ s . (56)Now from(57) x ss = x uu | x u | − (cid:104) x uu , x u (cid:105) x u | x u | we have that (cid:107) x ss (cid:107) ∞ is bounded provided | x u | is bounded away from zero for all u . As-suming this is the case we have furthermore from (56) that | (grad L x ) uu | is bounded andgrad L x ∈ C . We may therefore consider the flow as an ODE in Imm . Short time ex-istence requires a C Lipschitz estimate. One can estimate (cid:107) grad L x − grad L y (cid:107) C muchthe same as in Lemma 4.6. From (56) and product expansions as in Lemma 4.6: | (grad L x ) uu − (grad L y ) uu | ≤ | x uu − y uu | + c | | x u | − | y u | | + c | x ss − y ss | + c | G ( x ) − G ( y ) | + c | x uu − y uu | + c | x s − y s | . The result now follows from the Lipschitz estimate for G , together with estimates | x s − y s | ≤ c | x u − y u | and | x ss − y ss | ≤ c | x uu − y uu | which also follow from product expansions usingeg (57).It follows from (47) that if X (0) is C , then X ( t ) is C for all t < ∞ , and moreover | X u ( t ) | is bounded away from zero for all t < ∞ , so we have global existence for the C flow. Suppose X ( t ) is C for a short time, then from (56) ddt | X uu | = − | X uu | − | X u | (cid:28) X uu , (cid:90) X ˜ s ˜ s G d ˜ s (cid:29) + 2 (cid:104) X uu , X s (cid:105) (cid:28) X uu , (cid:90) X ˜ s G d ˜ s (cid:29) . From (57) notice | X ss | ≤ | X uu | | X u | − and therefore ddt | X uu | ≤ | X uu | ( c (cid:107) X uu (cid:107) ∞ + | X uu | )where c = (cid:107) X u (cid:107) ∞ sup u | X u | − . Supposing that at time t , (cid:107) X uu (cid:107) ∞ is attained at u , itfollows that ddt | X uu | ( u , t ) ≤ c | X uu | ( u , t )and therefore (cid:107) X uu ( u , t ) (cid:107) ≤ e ct . By the short time existence (cid:107) X uu (cid:107) ∞ is continuous in t ,so in fact (cid:107) X uu (cid:107) ∞ ≤ e ct and we have global C .For the Imm k case there is little that is novel and much that is tedious. Claim:(58) ∂ ku G = | X u | k ∂ ks G − (cid:104) ∂ ku X, X s (cid:105) G s + k − (cid:88) i P i ( X u , . . . , ∂ k − u X ) ∂ is G where each P i is polynomial in the derivatives of X up to order k −
1. From (55) this istrue for k = 2 . Assuming it is true for k we have ∂ ku G = | X u | k +1 ∂ k +1 s G + k | X u | k − (cid:104) X uu , X u (cid:105) ∂ ks G − (cid:104) ∂ k +1 u X, X s (cid:105) G s − (cid:104) ∂ ku X, X ss (cid:105)| X u | G s − (cid:104) ∂ ku X, X s (cid:105)| X u | G ss + ∂ u (cid:32) k − (cid:88) i P i ∂ is G (cid:33) = | X u | k +1 ∂ k +1 s G − (cid:104) ∂ k +1 u , X s (cid:105) G s + k (cid:88) i ˜ P i ∂ is G where each ˜ P i is polynomial in the derivatives of X up to order k . From (58) and ∂ ks G = − ∂ k ˜ s G we can calculate ∂ ku (grad L X ) = ∂ ku X + (cid:90) X∂ ks G d ˜ s = ∂ ku X + | X u | k (cid:90) ∂ k ˜ s XG d ˜ s − (cid:104) ∂ ks X, X s (cid:105) (cid:90) X ˜ s G d ˜ s + k − (cid:88) i P i (cid:90) ∂ i ˜ s XG d ˜ s (59)and observe that if X is in Imm k then so is grad L X . We may therefore consider thegradient flow as an ODE in Imm k . Short time existence requires a C k Lipschitz estimate.We claim that such an estimate can be proved inductively using (59) by similar methodsto those used above for the C case, except with longer product expansions. As it is thesame technique but only with a longer proof, we omit it.In summary, we have: Theorem 5.6.
Let k ∈ N be a natural number. For each X ∈ Imm k there exists aunique eternal H ( ds ) curve shortening flow X : S × R → R in C ( R ; Imm k ) such that X ( · ,
0) = X . N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 31 Curvature bound for the rescaled flow.
In this subsection, we study the H ( ds )curve shortening flow in the space of C immersions. This means that the flow has a well-defined notion of scalar curvature. Note that while the arguments in the previous sectionshow that the C -norm of X is bounded for all t , they do not show that this bound persiststhrough to the limit of the asymptotic profile Y ∞ . They need to be much stronger for thatto happen: not only uniform in t , but on X they must respect the rescaling factor.The main result in this section (Theorem 5.7) states that this is possible, and that thelimit Y ∞ of the asymptotic profile in the C -space enjoys C regularity, being an immersionwith bounded curvature.We start with the commutator of ∂ s and ∂ t along the flow X ( t, u ). Given a differentiablefunction f ( u, t ): f st = − (cid:104) X ut , X u (cid:105)| X u | f u + 1 | X u | f ut = f ts − (cid:104) X ts , X s (cid:105) f s . (60)From X ss = kN we have X sst = kN t + k t N and then using (cid:104) N t , N (cid:105) = 0, k t = (cid:104) X sst , N (cid:105) . Applying (60) twice X sst = X sts − (cid:104) X ts , X s (cid:105) X ss = ( X ts − (cid:104) X ts , X s (cid:105) X s ) s − (cid:104) X ts , X s (cid:105) X ss = X tss − (cid:104) X tss , T (cid:105) T − (cid:104) X ts , kN (cid:105) T − (cid:104) X ts , T (cid:105) kN and then k t = (cid:104) X tss , N (cid:105) − k (cid:104) X ts , T (cid:105) = −(cid:104) (grad L X ) ss , N (cid:105) + 2 k (cid:104) (grad L X ) s , T (cid:105) = − k − (cid:104) grad L X , N (cid:105) + 2 k (cid:104) T + T ∗ G, T (cid:105) = k − (cid:104) grad L X , N (cid:105) + 2 k (cid:104) T ∗ G, T (cid:105) where (grad L X ) ss − grad L X = kN (from (11)) and grad L s = T + T ∗ G have been used.Therefore(61) ddt k = 2 k − (cid:104) grad L X , kN (cid:105) + 4 k (cid:104) T ∗ G, T (cid:105)
Now letting ϕ ( t ) := k Y = e − t k using (21) and (34) to estimate (61), we find ϕ (cid:48) ( t ) ≤ e − t ( | k |L + k L ) ≤ e − t + 52 L ϕ ( t ) . Note that in the second inequality we used a ≤ a /
4, which holds for any a ∈ R .Integration gives | ϕ ( t ) | ≤ c + (cid:90) t c L | ϕ | dτ and so by the Bellman inequality ([25] Thm. 1.2.2) ϕ ( t ) ≤ ce (cid:82) t L dτ Since L ( X ) decays exponentially (42), we have that ϕ is uniformly bounded.This gives stronger convergence for Y in the case of C data, and we conclude thefollowing. (Note that the fact Y ∞ is an immersion followed already from (54).) Theorem 5.7.
Let X be an H ( ds ) curve shortening flow with X (0) ∈ Imm , and Y itsasymptotic profile. There is a non-trivial Y ∞ ∈ Imm such that Y ( t ) → Y ∞ in C as t → ∞ . That is, the asymptotic profile converges to a unique limit that is immersed withwell-defined curvature. Isoperimetric deficit.
In this section, we show that the isoperimetric deficit of thelimit of the asymptotic profile Y ∞ is bounded in terms of the isoperimetric profile of X .This is in a sense optimal, because of the great variety of limits for the rescaled flow, itis not reasonable to expect that the profile always improves. Indeed, numerical evidencesuggests that the profile is not monotone under the flow. Nevertheless, it is reasonable tohope that the flow does not move the isoperimetric deficit too far from that of the initialcurve, and that’s what the main result of this section confirms.5.6.1. Area.
We start by deriving the evolution of the signed enclosed area. Using (60)we find X st = X ts − (cid:104) X ts , X s (cid:105) X s = X ts − (cid:104) X ts , T (cid:105) T .
Differentiating (cid:104)
N, T (cid:105) = 0 and (cid:104)
N, N (cid:105) = 1 with respect to t yields (cid:104) N t , T (cid:105) = −(cid:104) N, X st (cid:105) , and (cid:104) N t , N (cid:105) = 0 . Therefore(62) N t = −(cid:104) N, X st (cid:105) T = −(cid:104) N, X ts (cid:105) T .
Using the area formula A = − (cid:82) L (cid:104) X, N (cid:105) ds , and ds = | X u | du implies ddt ds = (cid:104) X ts , X s (cid:105) ds ,we calculate the time evolution of area as dAdt = − (cid:90) L (cid:104) X t , N (cid:105) + (cid:104) X, N t (cid:105) + (cid:104) X, N (cid:105)(cid:104) X ts , X s (cid:105) ds = − (cid:90) L (cid:104) X t , N (cid:105) − (cid:104) X, T (cid:105)(cid:104)
N, X ts (cid:105) + (cid:104) X, N (cid:105)(cid:104) X ts , X s (cid:105) ds = − (cid:90) L (cid:104) X t , N (cid:105) + (cid:104) X ts , (cid:104) X, N (cid:105) T − (cid:104) X, T (cid:105) N (cid:105) ds . Now since ∂ s ( (cid:104) X, N (cid:105) T − (cid:104) X, T (cid:105) N ) = − N , integration by parts gives dAdt = − (cid:90) L (cid:104) X t , N (cid:105) ds . (63)5.6.2. Estimate for the deficit.
Consider the isoperimetric deficit D := L − πA . From ddt L = (cid:82) (cid:104) kN, grad L X (cid:105) ds and (63) we find ddt D = (cid:90) (2 L k − π ) (cid:104) N, grad L X (cid:105) ds . With the gradient in the formgrad L X = (cid:90) ( X (˜ s ) − X ( s )) G ( s, ˜ s ) d ˜ s we use the second order Taylor approximation G = −
12 sinh( L / (cid:18) | s − ˜ s | − L / + o ( L ) (cid:19) . N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 33 Note that (cid:82) X (˜ s ) − X ( s ) d ˜ s = L ( ¯ X − X ( s )) and moreover (cid:90) (2 L k − π ) (cid:104) N, L ( X − ¯ X ) (cid:105) ds = − LD where the ¯ X term vanishes because kN and N are both derivatives and L ¯ X is independentof s . Hence ddt D = 12 sinh( L / (cid:18) −LD (cid:18) L (cid:19) − (cid:90) (2 L k − π ) (cid:104) N, (cid:90) ( X (˜ s ) − X ( s )) (cid:18) | s − ˜ s | − L| s − ˜ s | + o ( L ) (cid:19) d ˜ s ds (cid:19) . For the terms involving k we have, for example, (cid:90) (cid:28) L kN, (cid:90) ( X (˜ s ) − X ( s ))( s − ˜ s ) d ˜ s (cid:29) ds = − (cid:90) (cid:28) L T, (cid:90) X (˜ s ) − X ( s ))( s − ˜ s ) − T ( s )( s − ˜ s ) d ˜ s (cid:29) ds and therefore we estimate ddt D ≤
12 sinh( L / (cid:18) −LD (cid:18) L (cid:19) + o ( L ) (cid:19) Because L L / ≤ D ≥
0, we have ddt
D ≤ − L L /
2) 2 D + o ( L ) . For the isoperimetric deficit D Y of the asymptotic profile Y , we have D Y = e t D , ddt D Y = e t ddt D + 2 D Y hence ddt D Y ≤ D Y (cid:18) − L L / (cid:19) + o ( L ) e t . From Lemma 5.1 we can take t ≥ t / such that o ( L ) e t decays like e − t for t > t / . If ≥ −(cid:107) X (0) (cid:107) ∞ L (0) we can find the required t / by solving = 1 −(cid:107) X (0) (cid:107) ∞ L (0) e − Ct / ,otherwise t / = 0. The constant C is from the gradient inequality and also depends on (cid:107) X (0) (cid:107) ∞ . Therefore the estimate for the integral of the extra terms depends only on X (0).Integrating with respect to t gives D Y ≤ D Y (0) c ( X (0)) e (cid:82) tt / − L L / dτ . The Taylor expansion for x (cid:55)→ x/ (2 sinh( x/ D Y ≤ D Y (0) c ( X (0)) e (cid:82) tt / L + o ( L ) dτ . Now using again the exponential decay of L we find D Y ≤ c ( X (0)) D X (0) . Summarising, we have:
Proposition 5.8.
Let X be an H ( ds ) curve shortening flow and Y its asymptotic pro-file. There is a non-trivial Y ∞ ∈ H ( S , R ) such that Y ( t ) → Y ∞ in C as t → ∞ .Furthermore, there is a constant c = c ( (cid:107) X (0) (cid:107) ∞ ) such that the isoperimetric deficit of Y ∞ satisfies D Y ∞ ≤ c D X (0) . A chord-length estimate and embeddedness.
In this section we prove that ifthe flow is sufficiently embedded (relative to total length) at any time, it must remainembedded for all future times. This holds also for the asymptotic profile.We achieve this via a study of the squared chord-arc ratio : φ := C h S where, writing s = s ( u ) , s = s ( u ), C h ( t ) := | X ( u , t ) − X ( u , t ) | and S := | s − s | . Recalling that X t = − X − (cid:90) XG d ˜ s = − X + ¯ X − (cid:90) ( X − ¯ X ) G d ˜ s we find ddt C h = − C h − (cid:28) X ( u ) − X ( u ) , (cid:90) ( X − ¯ X )( G ( s , s ) − G ( s , s )) d ˜ s (cid:29) and define Q := (cid:28) X ( u ) − X ( u ) , (cid:90) ( X − ¯ X )( G ( s , s ) − G ( s , s )) d ˜ s (cid:29) so that ˙ C h = −C h − C h Q . Assuming s > s we have S = (cid:82) s s ds and from ddt ds = (cid:104) X ts , X s (cid:105) ds we obtain˙ S = (cid:90) s s (cid:104)− T − T ∗ G, T (cid:105) ds = − (cid:90) s s ds − (cid:90) s s (cid:104) T ∗ G, T (cid:105) ds .
Now let Q := (cid:90) s s (cid:104) T ∗ G, T (cid:105) ds and then ˙ S = −S − Q . Therefore the time evolution of the squared chord-arc ratio is given by˙ φ = 2 C h ˙ C hS − C ˙ S S = 2 φ ˙ C h C h − φ ˙ SS = − φ Q C h + 2 φ Q S . Using the estimates (that follow via Poincar´e and (34)) | Q | ≤ L C h S| Q | ≤ L S N THE H ( ds )-GRADIENT FLOW FOR THE LENGTH FUNCTIONAL 35 and recalling the length decay estimate (42) we see that ddt φ = − φ Q C h + 2 φ Q S≥ −L (cid:112) φ (1 + (cid:112) φ ) . Therefore (cid:112) φ (cid:48) ≥ − L (cid:112) φ − L . Lemma 4.16, and choosing the appropriate ε in the proof of Lemma 4.15, implies that L ( t ) ≤ L (0) e − β ( X ) t where β ( X ) = 1 / (cid:112) (cid:107) X (cid:107) ∞ .Now let us impose the following hypothesis on X :(64) inf s ∈ [0 , L ] (cid:112) φ ( s ) > L β ( X ) e L β ( X . We calculate ddt (cid:18) e (cid:82) t L ( τ ) dτ (cid:112) φ (cid:19) ≥ e (cid:82) t L ( τ ) dτ (cid:16) − L (cid:17) ≥ − L e − β ( X ) t + (cid:82) t L e − β ( X τ dτ ≥ − L e − β ( X ) t + L β ( X = − L e L β ( X e − β ( X ) t . Integration gives e (cid:82) t L ( τ ) dτ (cid:112) φ ≥ (cid:112) φ − L β ( X ) e L β ( X . By hypothesis (64) the RHS is positive, and so the function √ φ can never vanish. Sincethe chord-arc length ratio is scale-invariant, the same is true for the asymptotic profile Y . Moreover, the hypothesis (64) may be satisfied simply by scaling any embedded initialdata (again, φ is scale-invariant, but the RHS of (64) is not). Thus we have the followingresult: Proposition 5.9.
Let X be an H ( ds ) curve shortening flow. Suppose X ∈ Imm satisfies inf s ∈ [0 , L ] C h ( s ) S ( s ) > L (cid:112) (cid:107) X (cid:107) ∞ e L √ (cid:107) X (cid:107) ∞ where, writing s = s ( u ) , s = s ( u ), C h ( t ) := | X ( u , t ) − X ( u , t ) | and S := | s − s | . Then X (as well as its asymptotic profile and limit Y ∞ ) is a family of embeddings.Proposition 5.9 together with Theorem 5.6 completes the proof of Theorem 1.4 fromthe introduction. Moreover Theorem 5.7, Proposition 5.8 and Proposition 5.9 completethe proof of Theorem 1.5. References [1] Ben Andrews and Paul Bryan. Curvature bound for curve shortening flow via distance comparison anda direct proof of grayson’s theorem.
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JSPS International Research Fellow, Mathematical Institute, Tohoku University, Sendai980-8578, Japan
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