Evolving convex curves

Transcription

1 Calc. Var. 7, c Springer-Verlag 1998 Evolving convex curves Ben Andrews Mathematics Department, Stanford University, Stanford CA 94305, USA Received May 1, 1997 / Accepted December 5, 1997 Abstract. We consider the behaviour of convex curves undergoing curvaturedriven motion. In particular we describe the long-term behaviour of solutions and properties of limiting shapes, and prove existence of unique solutions from singular or non-strictly convex initial curves, with sharp regularity estimates. 0. Introduction This paper is concerned with the motion of curves in the plane under curvaturedriven evolution equations. Here a moving curve is described by a family of embeddings x t : C R of a fixed curve C, and the evolution equations considered have the following form: 1 ẋ = Fn where ẋ = x t, and F = Fκ, n is a smooth function which prescribes the dependence of the speed on the curvature κ and the outward normal direction n. All curves considered are convex and embedded, so the function F is assumed to be defined only for non-negative κ. The main further restriction on F is that it be strictly increasing in κ, which amounts to the flow being parabolic. In many parts of the paper it is further assumed that F is homogeneous of some degree α/= 0 in the curvature: F = 1 α ψnκα, where ψ is a positive smooth function defined on S 1. Flows of this form will be called homogeneous of degree α. Other special cases of interest are isotropic Research partly supported by NSF grant DMS and a Terman fellowship. Mathematics Subject Classification: 35K15, 35K55, 35K65, 53A04 Present address: CMA, ANU, ACT 000 AUSTRALIA

2 316 B. Andrews flows with F independent of n, and symmetric flows with Fκ, n =Fκ, n for all κ>0 and all n S 1. Section I of the paper describes flows of the form with α<0, which expand the solution curves to infinite size. This takes finite time if α< 1, and infinite time if 0 >α 1. It is proved that a solution exists starting from any smooth, strictly convex initial curve, and that the resulting curves converge to a limiting shape if they are rescaled about the origin to keep their enclosed area constant. The limiting shape is independent of the initial curve, and is the unique closed convex curve up to scaling for which the motion described by Eq. 1 is simply homothetic expansion about the origin Theorem I1.1. Some attention is also given to the behaviour of the solutions at the initial time. The main result of this investigation is a proof of the existence of unique solutions starting from non-smooth or non-strictly convex initial curves Theorem I.1. The argument involves some new a priori estimates on the speed function which are proved by combining barrier arguments with a Harnack inequality. A large class of non-homogeneous expansion flows are treated in Sect. I3. The second main section of the paper is concerned with contraction flows with F positive, or α positive in the homogeneous case. In this case, any smooth, convex, embedded initial curve C 0 gives rise to a solution which remains smooth for a finite time and then converges to a single point Theorem II1.3. The behaviour of solutions at the final time is more complicated in these contracting cases, and is described in Theorem II1.11: For any positive smooth function ψ and any exponent α> 1 3, the solution curves converge for a subsequence of times to a limiting shape, after rescaling about the final point to keep the enclosed area fixed. As in the expanding case, the limiting shapes are such that their motion under Eq. 1 is simply homothetic scaling. Solutions also exist starting from singular convex initial curves Theorem II.8, and in such cases the regularity of solutions depends on the exponent α. We give sharp estimates on the level of regularity for each α the solution curves are C if α 1, but only C k+,γ, where k + γ =1/α 1, if α>1. However, this loss of regularity can only occur at points where the curvature is zero, and we give estimates on all higher derivatives in terms of κ 1 Theorems II1.9 and II.5. We expect that the subsequential convergence to the limiting shapes can be improved to C k+,γ convergence as the final time is approached. The difficulty in proving this is related to the fact that a given flow need not have a unique possible limiting shape: In [A6] we construct examples of flows with any given exponent α for which different initial curves may give rise to distinct limiting shapes. In a separate paper [A5] it is proved that the value α = 1 3 is sharp, in that flows with smaller values of α always have solutions which do not converge after rescaling. However we do prove here that if the isoperimetric ratio of the solution curves remains bounded, the rescaled curves converge smoothly to a homothetic limit Theorem II1.1. The special case α = 1 3 and ψ = 1 is the affine normal flow, which has been considered in [ST] and [A4]. In the case where the flow is symmetric, the results proved here are somewhat stronger: If the exponent α is 1 or greater, then there is only one possible limiting

3 Evolving convex curves 317 shape Corollary II3.. [A6] shows that the value 1 is optimal, by constructing examples of symmetric flows with any given α 1 3, 1 with several distinct homothetically shrinking solutions. The paper [A7] deals specifically with the isotropic homogeneous contraction flows, and gives a complete classification of all limiting shapes. Section II4 concerns non-homogeneous contraction flows, of a form allowing the existence of at least one homothetic solution. If the rate of growth of the speed as a function of the curvature is not too small, then there is only one such homothetic solution, and all other solutions converge to this Theorem II4.1. Many of the features described here the general dependence of behaviour on the degree of homogeneity of the speed, and the effects of anisotropy may be expected to carry over to curvature-driven parabolic evolution equations for convex hypersurfaces in higher dimensions. In particular, there are implications for the relative mean curvature evolution equations which have been proposed by Cahn, Handwerker and Taylor [CHT], and other models proposed by Angenent and Gurtin [AG] and others. The case of anisotropic Gauss curvature flows is discussed in [A8]. Some of the results presented here have also appeared elsewhere: In particular, some of the results on expansion flows appeared in the author s thesis [A1, Sect. III, chapter 1] in the isotropic case. Urbas [U1 ], Huisken [Hu4] and Gerhardt [Ge] have considered homogeneous equations with α = 1 inthe isotropic case. Their results apply in the more complicated setting of hypersurfaces in higher dimensions, and the papers [U] and [Ge] allowed convexity to be weakened to star-shapedness as long as the speed is positive. The methods used in the above papers are quite different to those presented here. The isotropic curve-shortening flow the contracting flow with α = 1 and ψ = 1 has been considered by many authors, most notably [Ga1 ], [GH], and [Gr]. Some of the results on contracting anisotropic flows in the symmetric case appeared in [A1] Section III, chapter, and have also been proved by Gage [Ga3] in the case of exponent equal to 1. Chow and Gulliver [CG] and Chow and Tsai [CT], [Ts] have used different methods to prove results similar to those given here for expanding flows, under the assumption of isotropy. The author would like to thank Tom Ilmanen, Sigurd Angenent, Rick Schoen, Leon Simon, Klaus Ecker, Gerhard Huisken and Alexander Isaev for useful ideas and discussions relating to various parts of the paper. I. Expansion flows This section examines expansion flow equations, under which curves move outwards with speed given by an increasing function of the radius of curvature. For convenience we modify the notation of the introduction for the purposes of this section, so that the evolution equation takes the following form:

4 318 B. Andrews t x = f κ, n n. In the homogeneous case we take β = α, so that f r, n = 1 β ψnr β. Section I1 deals with the homogeneous case with smooth, strictly convex initial curves. Section I allows initial curves which are merely convex, and not necessarily smooth or strictly convex. Section I3 contains some extensions to non-homogeneous flows. I1. Homogeneous flows with smooth initial curves Consider a family of convex curves given by a map x : C [0, T R comprised of embeddings x t = x., t for each t, and satisfying the equation 1.1 t x = 1 β ψnκ β n. ψ is a strictly positive, smooth function defined on the unit circle S 1, and the exponent β is positive. For each t denote by C t the curve x t C, and let A t be the area enclosed by C t. Theorem I1.1 Let x 0 : C R be a smooth, strictly convex embedding. Then there exists a unique solution x : C [0, T R of Eq. 1.1 with initial data x 0. The solution remains smooth and strictly convex on [0, T. If0 <β 1, then T = and lim t inf p C xp, t =. The rescaled curves given by the embeddings π A t xp, t converge in Hausdorff distance to a smooth, strictly convex limit Σ, and their support functions converge in C to the support function σ of Σ. Ifβ>1, then T < and as t approaches T, the inradius of C t converges to infinity. The rescaled curves converge in the Hausdorff metric to a limit Σ. The support functions of the curves converge in C, β 1 to the support function σ of Σ. If Σ has bounded curvature then it is smooth and strictly convex and the convergence takes place in C. In both cases, Σ is the unique embedded closed curve with enclosed area π which evolves under Eq. 1.1 by homothetically expanding from the origin. Remark. i. The support function of a convex curve is defined below. ii. In the case β>1the result presented here is the best possible, in that there are examples of smooth positive ψ for which the support function of the homothetic solution is only C, β 1. However there are several particular cases of interest where C convergence is attained: If ψ is constant, or has antipodal symmetry on S 1,orhask-fold symmetry for some k 3; or if ψ is given explicitly in the form σ where σ is the support function of some smooth, strictly convex r[σ] β curve enclosing the origin, and r[σ] is its radius of curvature. For given ψ, however, we know of no simple test which determines whether the homothetic limit has bounded curvature. Schauder estimates for parabolic equations imply

5 Evolving convex curves 319 that the support function and the convergence are C away from points where σ = 0. iii. In general the rescaled embeddings do not converge to a limiting embedding, except in the isotropic case. iv. For β>1 the limiting shape Σ may not be smooth, and the sense in which it evolves homothetically under the evolution equation is to be interpreted in the sense of Lemma I1. below: The support function σ of Σ has continuous second derivatives, and the solution of Eq. 1. with initial condition σ is proportional to σ at each time. Proof Theorem I1.1. A useful first step is to rewrite the evolution equation as a single scalar parabolic equation for a function on the unit circle. This can be achieved either by considering the evolving curves as graphs over a suitable circle, or by considering the support function. We adopt the latter approach which will also simplify many of the computations in later parts of the paper. The support function s : S 1 R of a convex embedded curve C 0 is defined by sz = sup x C0 x, z for each z S 1. Equivalently, if C 0 is given by an embedding x : C R then xp, np = snp for each p C. A detailed discussion of support functions can be found in [Sc]. We note that the curve is uniquely determined by its support function [Sc, Theorem 1.7.1] and that any function on S 1 whose homogeneous degree one extension to R is smooth and strictly convex in nonradial directions is the support function of some smooth, strictly convex curve [A, Eq..10]. The radius of curvature of the curve at the point with normal z is given by r[s]z := s z+sz where θ is the standard θ angle parameter on S 1 [A, Eq..11]. Lemma I1. Suppose x : C [0, T R is a solution of Eq Then the support functions s : S 1 [0, T R satisfy the equation 1. t sz, t = 1 β β ψz s z, t+sz, t. θ Conversely, if s : S 1 [0, T R is a smooth solution of Eq. 1. with r[s] > 0, and if x 0 : C R is a{ smooth embedding with image equal to the corresponding initial curve C 0 = s 0 zz + s 0 z θ z θ }, : z S 1 then there exists a unique solution x : C [0, T R to Eq. 1.1 which has initial data x 0 and such that the curve x t C has support function s t for each t [0, T. Proof. For a smooth, strictly convex solution x of Eq. 1.1, we have sz, t = xn 1 z, z where n 1 : S 1 C is the inverse of the Gauss map n. The evolution of s can then be calculated as follows:

6 30 B. Andrews t sz, t = t xn 1 z, z = t xn 1 z, z = t x n 1 z+tx t n 1 z, z = 1 β ψzκn 1 z β + Tx t n 1 z, z = 1 β ψzr[s]zβ since Tx t n 1 z is a vector tangential to C t, and since r[s]z =κn 1 z 1 as noted above. Conversely, if s : S 1 [0, T R is smooth, s t = s., t extends to a convex homogeneous degree one function on R for each t, and s satisfies Eq. 1. then the embeddings x : S 1 [0, T R defined by 1.3 xz, t =sz, tz + s z z, t θ θ define a set of curves {C t } which have support function s. These embeddings satisfy an evolution equation which can be determined as follows: t xz, t = t sz, tz + s z tθ θ = 1 β ψzr[s]zz + 1 z ψzr[s]z β θ θ = 1 β ψ n x z κ x z β + T xv where n x and κ x are the normal and curvature corresponding to the embedding x, and V TS 1 [0, T is the vector field on S 1 given by κ x β θ ψn x κ β z x θ. Here we have used the fact that T xv =κ 1 x V for any V TS 1. The equation that x satisfies is thus very similar to Eq. 1.1, differing only by the tangential term T xv. We now proceed to construct a family of diffeomorphisms φ : C [0, T S 1 such that xp, t = xφp, t, t gives a solution to Eq Take φp, 0 = n x0 p and solve the following ordinary differential equation for each p: d 1.4 φp, t = V φp, t, t. dt This equation has a unique solution for each p as long as s exists and remains smooth, and φ t is a diffeomorphism for each t. The family of embeddings x satisfies xp, 0 = x 0 n x0 p = x 0 p, and evolves according to the following equation:

7 Evolving convex curves 31 t xp, t = xφp, t, t t = t xφp, t, t+t x φp, t, t t = 1 β ψn x φp, t, t κ x φp, t, t β + T xv φp, t, t + T x V φp, t, t = 1 β ψn x p, tκ x p, t β where we have used that n x p, t =n x φp, t, t and κ x p, t =κ x φp, t, t. Hence x satisfies Eq. 1.1 as claimed. If ˆx is any other smooth solution of Eq. 1.1 giving rise to the same support functions s and agreeing with x 0 at the initial time, then for ˆx t and x x have the same image and hence are related by a diffeomorphism: ˆx t = x ˆφ t. The above calculation in reverse shows that ˆφ satisfies Eq. 1.4 with the same initial data as φ. By the uniqueness of solutions of the ordinary differential equation 1.4, ˆφ φ and ˆx x. Lemma I1. provides the basis for the proof of short-time existence and uniqueness in Theorem I1.1. Lemma I1.3 Under the conditions of Theorem I1.1 there exists a unique solution of Eq. 1.1 on some time interval [0, T. Proof. Existence follows from Lemma I1., since Eq. 1. is uniformly parabolic: Given x 0, there exists a unique smooth solution for some time to Eq. 1. with initial condition s 0 z = x 0 n 1 z, z. Then the second part of Lemma I1. gives a solution x of Eq. 1.1 beginning at x. Suppose there are two solutions x and ˆx of Eq Then by the first half of Lemma I1. they give rise to solutions s and ŝ of Eq. 1. with the same initial condition, and since the equation is strictly parabolic we have s = ŝ. But the second half of Lemma I1. says that there is a unique solution of Eq. 1.1 corresponding to the solution s of Eq. 1., so x =ˆx. Lemma I1.4 If C 0 has speed function f bounded below by a constant C 1 > 0, then this remains true throughout the interval of existence of the solution. Proof. The evolution equation for the speed function f is obtained by differentiating Eq. 1. with respect to t: sz, t = 1 [ ] t t β ψzβ r[s]z, tβ 1 r sz, t t 1.5 β 1 = ψr θ t s + ψr t β 1 s. This is a parabolic equation and the maximum principle applies to show that the minimum of t s is nondecreasing.

8 3 B. Andrews Lemma I1.5 For any smooth, strictly convex solution of Eq. 1.1 with support functions s t, the following integral quantity is nonincreasing: 1 Z := A 1 1 ψr[s] 1+β 1+β dθ, π S 1 where s is the support function corresponding to the solution. Furthermore Z strictly decreases unless the solutions are homothetic. The integral in Lemma I1.5 is often referred to as the entropy of the flow, because of an analogy between the case β = 1 and the entropy for the heat equation. These entropy estimates were first introduced in [GH] for the curve shortening flow β = 1 and ψ 1 and were also used by Gage and Li [GL] for anisotropic flows with the same homogeneity as the curve shortening flow. The extension to other degrees of homogeneity was carried out in [A3], by a proof similar to the one presented here, although the form of the equations is slightly different here in that we have not assumed the existence of a homothetic solution. Similar results hold for certain evolution equations for hypersurfaces in higher dimensions [Ch3, A3]. Proof. We use the Minkowski inequality see [Sc, Theorem 6..1] which states that for any two convex bodies K 1 and K in the plane with corresponding support functions s 1 and s, 1.6 s 1 r[s 1 ]dθ. s r[s ]dθ s 1 r[s ]dθ. S 1 S 1 S 1 We note that this inequality still holds if one of the functions say s 1 has r strictly positive, and the other is an arbitrary smooth function, since the transformation s s + Cs 1 leaves the difference of the two sides of Eq. 1.6 unchanged, and a choice of sufficiently large C makes r[s ] positive, so that s becomes the support function of some convex region. Also, the equality case of Eq. 1.6 occurs precisely when s z =cs 1 z+ z, e for some constant c and some point e R. Now calculate the evolution equation satisfied by the integral quantity Z from Lemma I1.5: β { t Z = Z 1+β ψr β r[ψr β ] dθ 1 } ψr β+1 dθ S 1 A S 1 0 βa 1+β since the bracket is just 1/A times the difference of the left and right hand sides of Eq. 1.6 in the case s 1 = s and s = ψr β : The enclosed area A is given by 1 S 1 sr[s] dθ. The equality case ocurs only when 1.7 sz =csz z, e t

9 Evolving convex curves 33 for some e R and some positive c. The reader may verify directly that if Eq. 1.7 is satisfied at some time t 0, then the unique solution of Eq. 1. is given by 1.8 sz, t = z, e + ρt sz, t 0 z, e, where ρt =1+1 βct t β for β/= 1 and ρt =e ct t 0 for β =1. The curves C t are the images of the embeddings x t given by Eq. 1.3, which are given by: 1.9 xz, t =e + ρt xz, t 0 e. Hence Eq. 1.7 is satisfied precisely when the solution curves evolve homothetically about the point e. Corollary I1.6 The isoperimetric ratio I = L 4πA remains uniformly bounded as long as the solution exists, with a bound depending on the initial value of the quantity Z in Lemma I1.5. Proof. First we note that a bound on Z for any flow of the form 1.1 implies a bound on the analogous quantity Z for the isotropic flow with the same degree of homogeneity: 1.10 Z = A 1 S 1 r[s] 1+β dθ 1 1+β 1 inf S 1 ψ Z. Now observe that a bound on Z implies a bound on the isoperimetric ratio, using the Hölder inequality: Z = 1 1 r[s] 1+β 1+β 1 dθ r dθ =π β A 1/ S 1 π β 1+β 1+β A 1/ S 1 Since Z is bounded by its initial value, the result follows. L 1/. We note that a bound on the isoperimetric ratio also gives a bound on the ratio of the circumradius the radius of the smallest ball which encloses the curve to the inradius the radius of the largest ball enclosed by the curve according to the Bonnesen inequality see for example [Sc], page 34. The next step in our argument is to bound the speed function f. This estimate differs in the two cases 0 <β 1 and β>1: Lemma I1.7 increases. If 0 <β 1, sup ψr β 1 β A s decreases and inf Proof. The evolution equation for A is computed as follows: A ψr β 1 β A s

10 34 B. Andrews t A = 1 sr[s] dθ t S = 1 ψr 1+β dθ. β S 1 It is useful to rewrite Eq. 1. in an alternative form: 1.1 s = ψrβ 1 t θ s+ψrβ 1 s + 1 β β ψrβ. Equations 1.11 and 1.1 combine with Eq. 1.5 to give an evolution equation 1 β for ψzrβ A s : 1.13 t fa 1 β s = A 1 β s 1 β fa s β 1 ψr θ f + ψrβ 1 f β+1 +1 β fa s β 1 = ψr θ +1 β ψr β 1 s θ + ψrβ 1 s +1 βf fa 1 β 1 β fa s s f r dθ S 1 + ψrβ 1 s S 1 f r dθ S 1 sr dθ f s s θ θ fa 1 β The case β = 1 follows directly from the parabolic maximum principle, since the first term is an elliptic operator, the second a gradient term, and the third 1 β vanishes. If 0 <β<1, and a maximum of fa s occurs at a point z 0, we have f z sz f z 0 sz 0 for all z S 1, and therefore f z f z 0 sz 0 sz. Multiplying by rz and integrating over z, we obtain f z 0 sz 0 S 1 f r dθ S 1 sr dθ, and hence from Eq and the parabolic maximum principle see for example 1 β [Ha4], Lemma 3.5 the maximum of fa s is nonincreasing. A similar argument works to show the minimum is nondecreasing. Next we obtain a speed bound for the case β>1: Lemma I1.8 Let β>1. If x satisfies Eq. 1.1, and sup C [0,t0 ] x R, then { } f f p, t 0 max R sup, Rβ sup S 1 ψ t=0 R s ββ 1 β. s

11 Evolving convex curves 35 Proof. As before, we choose the origin to be enclosed by the initial curve, so that f s is always positive. The quantity R s obeys the following evolution equation, which is a consequence of Eqs. 1.5 and 1.1: 1.14 f β 1 f = ψr t R s θ ψrβ 1 s f R s R s θ θ R s f R s β 1 R r β 1 ψr θ f R s f ψrβ 1 s f R s R s θ θ R s 1 sups 1 ψ β 1 f β β 1 R Rβ R s where we have used that r β β 1 sup S 1 ψ β ψrβ = since R s is no less than R. Hence if the maximum of R β sup S 1 ψ ββ 1 β β sup S 1 ψ f then it must be decreasing. This gives a bound on bound on f since R s R. f R s f R s Rβ f sup S 1 ψ R s is greater than and hence a We have now established that the curvature remains bounded above and below as long as the diameter of the solution remains bounded. Corollary I1.9 If T is the maximal time of existence of the solution, then diam C t as t T. Proof. The argument is similar to Theorem 14.1 of [Ha1] or Theorem 8.1 of [Hu1]: If the diameter remains bounded, then by Lemmas I1.7 and I1.8 the radius of curvature of the solution remains bounded on [0, T, and by Lemma I1.4 the radius of curvature remains bounded below. Therefore the evolution equation 1. is uniformly parabolic on [0, T, and consequently all higher derivatives of the support function remain bounded: C,α bounds follow from [K, Section 3.3] or from Lemma I1.1 below, and bounds on higher derivatives follow from Schauder theory. Hence the support functions s converge to a smooth limit s T as t approaches T. Since Eq. 1. is strictly parabolic, the solution can be extended for a short time beyond T, contradicting the assumption that T is the maximal time for which the solution exists. Lemma I1.10 T = if 0 <β 1, and T is finite if β>1. sup S 1 ψ Proof. For β =1,e β t sup t=0 s is a supersolution with bounded diameter for supt=0 t <. For 0 <β<1 a supersolution is s 1 1 β 1 β sup + S 1 ψ 1 β β t. The support function s of the solution is initially less than these supersolutions,

12 36 B. Andrews and hence it remains so, and is bounded for all finite times. If β > 1 then inf t=0 s β 1 β 1 inf S 1 ψ 1 β 1 β t is a subsolution which reaches infinite diameter in finite time. Since s is initially greater than this subsolution, s must also become infinite somewhere in finite time. We have proved the first part of Theorem I1.1. It remains to address the convergence behaviour of the rescaled solutions. An important step is to prove the existence of a homothetic solution. Lemma I1.11 There exists a homothetic solution Σ for the evolution equation 1.1. If 0 <β 1then Σ is smooth and strictly convex. If β>1then Σ is Lipschitz and has support function in C, β 1 S 1. Proof. In both cases we prove this by choosing good initial data for Eq. 1.1 for example a circle, and showing there is a subsequence of times for which the rescaled evolved curves converge to a homothetic limit. A uniform bound on the speed can be deduced from Lemma I1.7 or Lemma I1.8, together with the bound on the isoperimetric ratio from Lemma I1.6. We proceed to obtain a bound on the gradient of the speed on S 1 : Lemma I1.1 f { + f max θ sup S 1 {t} sup S 1 {0} f + f θ }, sup f. S 1 [0,t] Proof. f + f = f t θ θ θ = f θ θ ḟ r[f ] +fḟr[f ]. ḟ r[f ] +f ḟ r[f ] f θ ḟ r[f ] At a maximum of this quantity the derivative is zero, so we have f 0. If the maximum is so large that f θ θ f + f = θ f θ + f > sup t f, then we also have /= 0, and therefore r[f ] = 0. In the above evolution equation the first term is nonpositive at a maximum, and the other two terms vanish, so that the time derivative is non-positive, and the maximum is non-increasing. Proof Lemma I1.11, contd.. Lemma I1.1 shows that the speed function satisfies a Lipschitz bound; equivalently, r[s] β is Lipschitz, so r[f ]isinc 0, β 1, and the support function is in C, β 1. Hence as before we have a subsequence which

13 Evolving convex curves 37 converges to a C, β 1 function which has a zero time-derivative for the entropy, and hence is the support function of a homothetic solution of Eq Lemma I1.13 There is precisely one homothetic solution Σ of Eq. 1.1 with enclosed area π. Let σ : S 1 R be the support function of Σ. For any convex 1. solution of the evolution equation 1.1, let I = 4πA[C t ] C σ ds t Then I 1+ C A[C t ]. Consequently the ratio of the circumradius RC t,σ the smallest factor by which Σ can be scaled while still enclosing a translate of C t to the inradius rc t,σ the largest factor by which Σ can be scaled while remaining enclosed by some translate of C t converges to 1 as t approaches T. Proof. σ satisfies the following equation see Eqs : 1.15 ψzr[σ] β = C βσ. Hence ψ can be written in terms of σ, and the evolution equation 1. becomes β r[s] 1.16 t s = C σ. r[σ] Now we proceed to calculate the evolution equation for the isoperimetric difference for the unrescaled solution: If V 1 = S 1 σr[s]dθ, then 1.17 d [ V dt 1 4πA ] [ ] = σr[s]dθ f r[σ]dθ σr[σ]dθ f r[s]dθ S 1 S 1 S 1 S 1 where we have used the formula 1.11 for the enclosed area, and the fact that Σ has enclosed area π, as well as the identity S 1 sr[σ]dθ = S 1 σr[s]dθ which follows by integrating by parts twice. Now we use the form of f and Eq together with the Hölder inequality: ψ f r[s]dθ σr[σ]dθ = S 1 S 1 S 1 β r[s]β r[s]dθ σr[σ]dθ S 1 β+1 r[s] = C σr[σ] dθ σr[σ]d θ σ>0 r[σ] σ>0 β r[s] r[s] C σr[σ] dθ σr[σ] dθ σ>0 r[σ] σ>0 r[σ] = f r[σ]dθ σr[s]dθ σ>0 S 1 = f r[σ]dθ σr[s]dθ S 1 S 1 Thus the right-hand side of Eq is non-positive, and

14 38 B. Andrews sr[σ]dθ 4πA C, S 1 where C is the initial value of the left hand side. Dividing by 4πA we deduce RCt,Σ the required estimate. The conclusion regarding the ratio rc t,σ follows from the Diskant inequality see [Sc, Theorem 6..3]. Lemma I1.14 Let ςz, t = sz,t sup ςz,t σz,t.ifβ =1, then inf ςz,t then sup ςz, t inf ςz, t sup ςz, 0 1 β +1 βt inf ςz, 0 1 β +1 βt sup ςz,0 inf ςz,0.ifβ 0, β. Proof. ς lies above the constant function inf ςz, 0 initially, and below the constant function sup ςz, 0, and evolves according to the equation ς t = 1 β r = 1 β r[σ] β σ r[σ] ς + ς θ r[σ] θ β ς θ + ς By the maximum principle, ς is between inf ςz, 0 1 β +1 βt 1/1 β and sup ςz, 0 1 β +1 βt 1/1 β. Corollary I1.15 Under the hypotheses of Theorem I1.1, the rescaled solution converges in Hausdorff distance to the limit curve Σ, and the support function converges in the sense stated in Theorem I1.1. Proof. If 0 <β 1 then Lemma I1.14 gives Hausdorff convergence, and the speed bound of Lemma I1.7 and Schauder estimates give convergence of higher derivatives. If β>1 then by Lemma I1.13 the support functions s of the rescaled curves converge to σ up to translations: s t σ p t, z 0 uniformly for some p t R. The speed bound of Lemma I1.8 and the derivative bound of Lemma I1.1 imply that the curvatures of the rescaled curves converge in C 1/β. It remains to show that p t converges to zero, which we prove in the following Lemmata. Lemma I1.16 The quantity Q [st] = A[st] 1+β β decreases, strictly unless s = σ. S 1 ψ 1/β s 1+ 1 β dθ Remark. The case β = 1 of this result was proved in the work of Firey [Fi] on the flow of hypersurfaces by their Gauss curvature. Proof. Note that Q [st] = Q [ st], and A[ st] = π. Since

15 Evolving convex curves 39 t s = β 1 A ψr[ s] β 1 ψr[ s] 1+β dθ s, π π S 1 we have writing Rt = A[C t ]/π d ψ 1 β s 1+β β dθ dt S 1 = 1+β { βr 1 β ψ 1 1/β s 1/β r[ s] β dθ S 1 1 } ψr[ s] 1+β dθ ψ 1/β s 1+1/β dθ π S 1 S { 1 } 1+β = πβr 1 β ξ 1/β dµ ξ dµ ξ 1+1/β dµ dµ S 1 S 1 S 1 S 1 0 by the Hölder inequality, where dµ = ψ 1/β s 1+1/β dθ and ξ = ψ s 1 r[ s] β. Equality holds if and only if ξ is constant, which means s is the support function of a homothetic solution about the origin, and hence s = σ. Note that in this calculation we assume that the initial curve encloses the origin, so that s is always positive. Corollary I1.17 p t converges to zero as the final time is approached. Proof. Suppose this is not the case. Then there exists a sequence of times approaching T such that pt ε > 0. Assume without loss of generality that the initial curve encloses the origin, so that s is always positive. A[C As before write Rt = t ] π. Define for each τ [0, T and each t [ Rτ β 1, Rτ β 1 T τ, Then ŝθ, t,τ= 1 Rτ s θ, τ + Rτ β 1 t t ŝ = ψr[ŝ]β. The entropy estimate of Lemma I1.5 gives d dt A = 1 β S 1 ψr 1+β dθ π β Z01+β A 1+β, so that d dt A 1 β πβ 1 Z0 1+β. β Integrating from τ to T and noting that lim t T A =, we obtain

16 330 B. Andrews T τrτ β 1 t 0 = β β 1πZ0 1+β. Therefore the solution ŝ is well-defined on S 1 [0, t 0 /] [0, T. By the same argument, for each t the enclosed area of the curve with support function ŝ is bounded, independent of τ and t in this range. It follows since the origin is enclosed that the support function ŝθ, t,τ is bounded. The curvature bound of Lemma I1.8 and the speed gradient bound of Lemma I1.1 imply that ŝ., t,τis uniformly bounded in C,1/β. Since the speed is bounded, the Lipschitz constant of ŝθ,., τ is bounded, independent of τ and θ. Note that ŝ., 0,τ = sτ. Therefore there is a subsequence of times τ k approaching T, such that the functions ŝθ, t,τ k ons 1 [0, t 0 /] {τ k } converge in C,γ in θ for any γ<1/β and C 0,1 in t to a limiting function ŝθ, t, T which is C,1/β in θ and C 1 in t and satisfies Eq Furthermore, from Lemma I1.13 and the assumption that pτ does not converge to zero we can assume that ŝθ, 0, T =σθ+ p, z, for some non-zero p R. By uniqueness of the solutions of Eq. 1.19, we have that ŝθ, t, T = 1 β 1 β t 1 β 1 σ + p, z for all t [0, t 0 /]. By Lemma I1.16, Q [ŝ., t, T ] is strictly decreasing in t, soq [ŝ., 0, T ] Q [ŝ., t 0 /, T ] = ε>0. Since Q [ŝ., t,τ k ] converges to Q [ŝ., t, T ], we can choose k 0 sufficiently large so that for all k k 0, we have Q [ŝ., 0,τ k ] Q [ŝ., 0, T ] 1 3 ε and Q [ŝ., t 0 /,τ k ] Q [ŝ., t 0 /, T ] ε. Passing to a subsequence if necessary, assume that τ k+1 >τ k + Rτ k β 1 t 0 / for all k k 0. Then since Q is decreasing, 0 Q [s., τ k+1 ] Q [s., τ k + Rτ k β 1 t 0 /] = Q [ŝ., 0,τ k+1 ] Q [ŝ., t 0 /,τ k ] Q [ŝ., 0, T ] 1 3 ε Q [ŝ., t 0/, T 1 3 ε ε 3 ε 1 3 ε, a contradiction. This completes the proof of Theorem I1.1.

17 Evolving convex curves 331 We conclude this section by mentioning some examples which demonstrate that the result of Theorem I1.1 is optimal. Proposition I1.18 For any β>1there exists a smooth, strictly positive function ψ : S 1 R such that the homothetic solution of Eq. 1.1 does not have bounded curvature, and is a Lipschitz curve with support function in C, β 1 but not in C,γ for any γ> 1 β. Proof. We choose the homothetic curve Σ and show that the resulting function ψ is smooth. Note that a curve is homothetic precisely when its support function σ satisfies the equation σ = ψr[σ] β, or equivalently 1.0 θ σ + σ = ψ 1 β σ 1 β. In the case ψ = 1 this ordinary differential equation has solutions given by inverting the equation σ 1.1 θσ =θ dr. β 1+β r 1+β β r Now choose σ 0 on some interval [0,θ 0 ] with θ 0 <π. Continue on either side of this interval a short distance in S 1 by gluing in the solution from Eq This gives the support function of a convex curve which has a corner at the origin with angle θ 0 and continues for some short distance with radius of curvature bounded. A similar gluing can be used to extend the curve in the other direction, and the resulting curve can be closed up smoothly by pasting in a smooth, strictly convex curve, and we take σ to be the support function of this curve. Then σ satisfies Eq. 1.0 with ψ 1 on a region slightly larger than [0,θ 0 ]ins 1, and ψ = σ which is smooth on S 1 \[0,θ r[σ] β 0 ]. The curve then evolves homothetically from the corner under Eq Such solutions cannot arise when ψ is antipodally symmetric, since then the homothetic solution must be antipodally symmetric since it is unique. Hence in these cases and in particular for isotropic flows Theorem I1.1 can be strengthened to show that inf C x converges to infinity, and that the limiting curve is smooth and is attained smoothly. The same conclusion follows as long as the homothetic solution for the flow has σ strictly positive. This follows from the results presented in Sect. I. I. Singular initial curves In this section we apply expansion equations to curves which are convex, but not necessarily smooth or strictly convex. The main result is the following:

18 33 B. Andrews Theorem I.1 Suppose C 0 is a curve given by the boundary of an open convex region in R. Then there exists a unique family of convex curves C t for 0 < t < T converging to C 0 in the Hausdorff distance as t approaches zero, with support functions sθ, t satisfying Eq. 1.. If 0 <β 1 then C t is smooth and strictly convex for all positive times, and the behaviour for large t is described by Theorem I1.1. If β>1 then the curves have C,1/β support functions for positive times, and expand to infinite inradius, converging in C,γ for any γ<1/β after rescaling to the unique homothetically expanding solution Σ of the flow. In the latter case the solution curves are strictly convex but may not be smooth: Specifically, any corner in C 0 persists for some positive time. If Σ has bounded curvature then the curves C t become strictly convex and smooth before the end of the interval of existence. Remark In the case β>1the support function is a classical i.e C, β 1 solution of Eq. I1. for positive times. The curves will in general be singular, however: The Hölder estimate on the second derivatives of the support function does not imply any regularity of the curve itself unless the curvature is also bounded. In particular, a convex curve with corners may have a smooth support function by corner we mean a point of the curve where the normal has a jump discontinuity. The resulting family of curves forms a viscosity solution of Eq. I1.1, but not in general a classical solution. In the case β = 1 the equation is linear and the result follows from the theory of weak solutions of linear heat equations. We will be concerned only with the proof of the case β/=1. Proof. We approach the proof of Theorem I.1 through the following series of a priori estimates: Lemma I. For any smooth, strictly convex solution of Eq. I1.1, with support function s, the total change in the support function is bounded as follows on an interval 0,τ where τ depends only on diamc 0 and sup S 1 ψ:.1 supsz, t sz, 0 C diamc 0, sup ψt S 1 S 1 The speed of motion f is also bounded for positive times:. sup S 1 f z, t C diamc 0, sup S 1 ψt β 1+β. 1 1+β. Proof Lemma I.. We begin by proving the estimate.1. The idea is to consider expanding spheres as barriers to bound the total distance travelled. Note that a sphere S expanding according to the equation.3 t x = 1 β sup ψκ β S 1

19 Evolving convex curves 333 is a supersolution of Eq. I1.1: Any surface evolving under Eq. I1.1 and initially enclosed by S remains enclosed by S as long as both solutions exist. This follows from the maximum principle since at a point where the two surfaces first meet, the curvature of the inside surface is greater than that of the outer one, and so its speed of motion is less. Choose a direction z in S 1. Since d = diamc 0 is bounded, a sphere of any sufficiently large radius at least d can be chosen to enclose C 0 and to have support function in direction z not exceeding that of C 0 by more than R R d. Now let these spheres evolve by Eq..3. The sphere at time 1 t has radius R t = R 1 β + 1 β β sup S 1 ψt 1 β.ifβ>1this expression is only Rβ valid for t < β 1 sup S 1 ψ. This shows that the difference between the support functions of C t and C 0 in direction z is no greater than [ R 1+ 1 β sup S 1 ψt 1 ] 1 β 1 d βr 1 β 1 for any R d, if0< β < 1, or any R d, β 1 β sup S 1 ψt β 1 if β>1. We now choose a good value of R to estimate the total change in { sz: Take R = t 1+β 1 1+β 1 for t max d interval we have 1+ 1 β sup 1 S 1 ψt 1 β βr 1 β 1 β sup S 1 ψt βr 1 1 β and d R 1+CtR β 1 =1+Ct, 1 R β 1 β sup S 1 ψ 1+β and 1+β }. On this. Combining these we have 1 d 1 C d = R R Ct 1+β. Hence on this time interval the total change in the support function in 1 direction z is no greater than Ct 1+β. To prove the estimate. on the speed, we need the following regularising effect: Lemma I.3 For any smooth solution of Eq. I1. the following estimate holds: β t β 1 f 0 if β>1, and t t β 1 β f 0 if 0 <β<1. t This estimate was proved in [A] by a different method, and comprises the main step in proving the Harnack estimate for these flows. I give a short proof here which illustrates the fact that these estimates depend only on the parabolic nature of the equations and the homogeneity of degree not equal to 1. Proof Lemma I.3. The evolution equation for the speed f can be rewritten as follows:

20 334 B. Andrews.4 f β 1 = 1 t β ψ β 1 r[f ]. Equation.4 inherits a homogeneity property from the evolution equation I1.: If f satisfies Eq..4, then the function f λ given by f λ z, t =λf z,λ β 1 β t is also a solution for any positive λ. Equation.4 also has a comparison principle, so that if f >gat time 0 and f and g both evolve according to Eq..4 then f >gas long as both solutions exist. In particular we have f λ > f for λ>1at time 0, and hence for all times. Thus λ f λ z, t 0. Explicitly differentiating the expression for f λ at λ = 1 gives β 1.5 f z, t+ t f z, t 0, β t which may be rewritten as t f β 1 β t 0. Proof Lemma I., continued. If 0 <β<1, choose t 0,τ] and z S 1. Then t t/ f z,ɛdɛ t 0 f z,ɛdɛ = sz, t sz, 0 Ct 1 1+β. Therefore there exists t 1 t/, t such that f z, t 1 Ct β 1+β. By Lemma I.3, ft β 1 β is β t 1 β nonincreasing, so f z, t f z, t 1 t 1 β 1 Ct β 1+β. 1 If β>1, choose t 0,τ/]. Then t f z,ɛdɛ t 1 f z,ɛdɛ C t 1+β, t 0 so there is t 1 t, t with f z, t 1 1+β 1 t β β 1+β. Since ft β 1 is nondecreasing, f z, t 1+β 1 t β t1 β β β t β β 1 Ct β 1+β. Next we control the gradient of the speed: Lemma I.4 For any smooth solution of Eq. I1., the following estimate holds: sup f z, t S 1 θ sup f z, t+ S 1

21

22 336 B. Andrews Lemma I.7 Under the hypotheses of Theorem I.1 there is at most one solution of Eq. I1. with initial curve C 0. Proof. We consider the two cases 0 <β<1 and β>1separately. If β>1, suppose s 1 and s are two solutions of Eq. I1.. Then t s z, t s 1 z, t = 1 β ψ r[s ] β r[s 1 ] β = 1 β ψrβ 1 r[s s 1 ] = 1 β ψrβ 1 θ s s β ψrβ 1 s s 1 where r lies between r[s ] and r[s 1 ], and therefore is bounded by Ct 1 β+1 for some C. By the maximum principle, t sup s s 1 Ct β 1 β+1 sup s s 1 S 1 S 1 and therefore s z, t s 1 z, t C sup S 1 s z,δ s 1 z,δ β 1 t ɛ β+1 dɛ. The δ integral here converges to a finite limit as δ approaches zero, and if the two solutions s and s 1 converge in Hausdorff distance to the same initial curve C 0 then sup S 1 s z,δ s 1 z,δ converges to zero as δ converges to zero. Hence s s 1 0 as long as both solutions exist. If 0 <β<1, and s and s 1 are two solutions of Eq. I1.1 converging in Hausdorff distance to the same initial curve C 0, then consider the following integral estimate: d s s 1 r s s 1 dθ = f f 1 r[s ] r[s 1 ] dθ dt S 1 S 1 = 1 ψ r[s ] β r[s 1 ] β r[s ] r[s 1 ] dθ β S 1 = r[s ] r[s 1 ] dθ Ct 1 β 1+β r[s ] r[s 1 ] dθ S 1 Now let w = s s 1 and integrate over t from 0 to some later time τ, observing that if s s 1 approaches zero in L norm as t approaches zero, and s and s 1 are both support functions of convex regions in the plane, then s s 1 approaches zero in the C 0,1 norm, and S 1 s s 1 r[s s 1 ]dθ approaches zero also. τ wr[w]dθ C t β 1+β r[w] dθdt. S 1 {τ} ψr 1 β S 1 0 S 1 {t} Note that the bounds on distance travelled from Lemma I. imply that the left hand side is bounded for each t > 0. Now multiply by an positive L 1 function

23 Evolving convex curves 337 gτ and integrate in τ from 0 to a time t 0, and then use Fubini s theorem to reverse the order of integration: t0 t0 τ wr[w]dθdτ C gτt β 1+β r[w] dθdtdτ 0 0 S 1 {τ} This yields the following: t0 Ct β t0 1+β gτdτ t = C = C 0 0 t0 t0 S 1 {t} 0 t S 1 {t} t0 t0 0 S 1 {t} { If we take gt =Bt β 1+β exp M 1 + βt t gτdτ gτt β 1+β r[w] dθdτdt t β 1+β S 1 {t} r[w] dθdt r[w] dθ gt wr[w]d θ dt 0. S 1 {t} 1 1+β } then gt =Mt β t 1+β 0 gτdτ t if we choose B appropriately. M is an arbitrary positive constant. Choosing M = C / we obtain t0 0 gt r[w] wr[w]dθdt 0. S 1 {t} However from the Wirtinger inequality on S 1 we have r[f ] f f f r[f ]dθ = S 1 S 1 θ dθ 0, θ and hence we have t0 gt r[w] dθdt 0, 0 S 1 {t} and so r[w] 0 for 0 < t t 0. Hence t w 0 on this interval, and w 0. This completes the uniqueness proof. The proof of Theorem I.1 can now be completed: We have shown the existence of a unique solution in C, β 1 for Eq. I1., and it remains to establish the long-term behaviour of these solutions. First consider the case 0 <β<1: We establish an a priori bound from below for the speed at any positive time, thus showing that the evolving curve immediately becomes smooth: Lemma I.8 If 0 <β<1 and s is a smooth solution of Eq. I1., then inf S 1 {t} f z, t min{c 1, C t 1 1 β }, where C depends on inf S 1 ψ, β, I [C 0 ] and diamc 0.

24 338 B. Andrews Proof. The proof uses comparison with small enclosed circles close to the initial curve. Since we have a bound on the initial isoperimetric ratio, we can place concentric circles S + and S such that S is enclosed by the initial curve C 0, and S + encloses the initial curve C 0, and such that the ratio of the radii of S + and S is bounded by a constant depending only on I see [Sc, Theorem 6..3]. For convenience we translate C 0 so that these circles are centred at the origin. Fix z S 1, and consider the convex hull Γ formed from the point x 0 z on C 0 with normal direction z together with the inner circle S. This is certainly enclosed by C 0. The circles S r, t = 1 rρ0 x 0 z+ r 1 β + 1 β 1 inf β ψt 1 β S 1 S 1 are contained in Γ for each r 0,ρ, and hence are enclosed by C 0 for t =0. The families of curves S r,. are subsolutions of Eq. I1., since they satisfy the evolution equation t s = 1 β inf ψr[s]β. S 1 They therefore remain enclosed by the evolved curve C t for each positive time. Hence sz, t = = sup y, z y convc t sup y, z y S r,t 1 r ρ z, x 0 z + r 1 β + 1 β β r 1 β + 1 β inf β ψt S 1 1 inf ψt 1 β S β, sz, 0 r ρ + + ρ for any r 0,ρ, where we have used the fact sz, 0 ρ +. This gives a bound below on the total change in s:.8 sz, t sz, 0 r 1 β + 1 β 1 inf β ψt 1 β ρ + r. S 1 ρ 1 1 β 1 β inf With r = min ρ, S 1 ψ 1 1 β t 1 β this yields: β ρ+ ρ 1.9 sz, t sz, 0 ρ min ρ, 1 β inf S 1 ψ ρ + 1 β β ρ+ ρ β t 1 1 β.

25 Evolving convex curves 339 This estimate may be combined with Lemma I.3 to yield the desired estimate in exactly the same manner as in the proof of Lemma I.. Corollary I.9 If 0 <β<1 the solution s of Eq. I1. is C for t > 0. Proof. We have bounds above and below on the speed for t > 0. Therefore Eq. I1. is uniformly parabolic, and so the solution s is smooth with r[s] > 0. By Theorem I1.3 the curve is also smooth and strictly convex. In the case β>1, there exist curves which evolve homothetically but are not smooth, so we cannot expect r[s] > 0 for t > 0. Indeed, we have: Lemma I.10 Fix β>1.ifc 0 has a corner i.e. r[s] =0on an interval [θ 0,θ 0 +ɛ] for some ɛ<π, then C t has a corner for all t t 0, where t 0 depends only on ψ, β, ɛ, and diamc 0. Proof. Given ɛ and θ 0, we can find a curve Σ which evolves homothetically under Eq. I1. with the same β and some anisotropy function ψ, and which has a corner on the interval of directions [θ 0 π ɛ 4,θ 0 + π+3ɛ 4 ]ins 1. sup ψ S 1 and inf ψ S 1 depend only on ɛ and β see the proof of Lemma I1.18. Σ contains a sector of a circle of some radius ρɛ, β between the angles θ 0 π/ and θ 0 + ɛ + π/. Then t σ = 1 β ψr[σ] β inf ψ S 1 1 sup S 1 ψ β ψr[σ]β Hence σ z, sup S 1 ψ inf S 1 ψ t is a supersolution of Eq. I1. with anisotropy function ψ. Place C 0 with the corner at the origin. Then C 0 is contained in the sector of the circle of radius diamc 0 between the angles θ 0 π/ and θ 0 + ɛ + π/, and therefore is contained in diamc 0 ρ Σ. By the comparison principle C t remains inside a larger multiple of Σ for each larger t, at least until this homothetic solution reaches infinite size. This lifetime depends only on ɛ, β, and diamc 0.In this time the homothetic solution is always contained in the cone of directions from θ 0 3π/4 +ɛ/4 toθ 0 +3π/4 +3ɛ/4, and hence the curve C t still has a corner of at least this angle for this time interval. In the case where the homothetic solution of the flow has support function bounded from below, we can show that all corners must disappear before the curve expands to infinite size, and hence that the evolving curves become smooth before the final time: Lemma I1.13 shows that a subsequence of times can be found on which the time derivative of the isoperimetric ratio of the evolving curve with respect to Σ becomes small. We have uniform bounds on the rescaled support function s and its second derivatives in some Hölder norm, and hence

26 340 B. Andrews in the limit we have convergence of s in C to a limiting function for which the time derivative of the I is zero. By the proof of Lemma I1.13, the time derivative of the isoperimetric ratio can only be equal to zero if the curve is a scaled translate of Σ. Since σ = C ψr[σ] β is positive, r[σ] is positive. But r[ s] converges to r[σ] and therefore is positive for t sufficiently large. Lemma I1.4 shows that it remains so, and Lemma I. shows that the speed is uniformly bounded from above. But then Eq. I1. is uniformly parabolic, and the solution is smooth. The conclusion follows from Theorem I1.1. I3. Nonhomogeneous expansion flows In this section some of the results of the previous two sections are extended to flows which are not homogeneous. We prove convergence to a homothetic solution by controlling the isoperimetric ratio. In the special case of isotropic flows, similar results were proved in [CT] using a reflection argument to control the oscillation of the support function. We consider a class of flows which are designed to have homothetic solutions: The speed of motion f in Eq. 0.1 is taken to be σf r[s] r[σ] where σ>0 is the support function of a smooth, strictly convex curve with enclosed area π, and F : R + R + has derivative everywhere positive. Thus σ evolves homothetically: sz, t =G 1 tσz satisfies Eq. 0.1 for t < 1 1 Fρ dρ, where Gr = r 1 1 Fρ dρ. The main theorem of this section is the following: Theorem I3.1 Let C 0 be a smooth, strictly convex embedded curve in R, given by an embedding x 0 : C R. Then there exists a unique smooth solution x of 3.1 t x = σf r[σ]κ 1 n. x., 0 = x 0. on a maximal time interval [0, T. T = if and only if dρ 1 Fρ =. x uniformly as t T, and π/ac t Σ in Hausdorff distance as t T. The key estimate is a generalisation of Lemma I1.13: Lemma I3. For any smooth strictly convex solution of Eq. 3.1, d A[Ct ]I 1 0, dt with equality if and only if C is a scaled translate of Σ. Proof. The integrals V 1 = S 1 σr[s]dθ and A evolve as follows:

27 Evolving convex curves 341 d r dt V 1 = σr[σ]f dθ; S 1 r[σ] and d r r dt A = σr[σ] F dθ. S 1 r[σ] r[σ] Combining these, we obtain an expression for the time derivative of V1 4πA: d 3. V dt 1 4πA = ξ dµ Fξ dµ dµ ξfξ dµ, S 1 S 1 S 1 S 1 where dµ = σr[σ]dθ and ξ = r/r[σ]. Then we use the following generalised Hölder inequality: Lemma I3.3 Let M be a compact manifold with a volume form dµ, and let ξ be a continuous function on M. Then for any non-decreasing function F, ξfξ dµ M M ξ dµ Fξ dµ M M dµ. If F is strictly increasing, then equality holds if and only if ξ is constant. Proof. By Fubini we can write ξ dµ Fξ dµ dµ ξfξ dµ M M M M = ξφfξθ ξφfξφ dµθdµφ M M = 1 ξφ ξθ Fξφ Fξθ dµθdµφ M M = ξφ ξθ Fξφ Fξθ dµθdµφ. ξφ>ξθ Since F is non-decreasing, the result follows directly from the last identity. This completes the proof of Lemma I3., since V 1 4πA =4πAI 1. Corollary I3.4 The only homothetic solutions of Eq. 3.1 are the curves with support functions equal to a multiple of σ. Proof. A homothetic solution must have constant isoperimetric ratio, and increasing enclosed area, and so V1 4πA =4πAI 1 unless I = 1. Since AI 1 is bounded, we must have I = 1 and so the solution is a scaled translate of Σ. But a translated multiple of Σ does not evolve homothetically about the origin, so the solution must be a multiple of Σ. Lemma I3.5 The minimum of the speed σf does not decrease.

28 34 B. Andrews Proof. The evolution equation for the speed is t σf = σ r[σ] F σf θθ + At a minimum of σf, both terms are non-negative. σ r[σ] F σf. Proposition I3.6 For any solution of Eq. 3.1, the following estimate holds: { } { sup σf θ +σf } { max sup σf θ +σf }, sup σf. S 1 {t} S 1 {0} S 1 [0,t] Proof. The proof of this is identical to that of Theorem I1.1. Corollary I3.7 If lim x Fx =, then { sup σf max sup { σf θ +σf }, S 1 {t} S 1 {0} If lim x Fx =F max <, then r r[σ] sup S 1 {t} sup σf 3V 1 π infσr[σ] r sup + tf max. S 1 {0} r[σ] }. In either case, the curvature is bounded as long as the inradius of the curve remains bounded. Proof. First consider the case lim x Fx =. By Proposition I3.6, either sup S 1 {t} { σf θ +σf } sup S 1 {0} { σf θ +σf } in which case sup σf S 1 {t} sup S 1 {0} { σf θ +σf }, or we have σf θ +σf supσf. [0,t] In the latter case, first suppose sup t σf = sup [0,t] σf. Then σf θ +σf supσf. t and by integrating from any point where the maximum of σf is attained, σf supσf cos θ