Evolving convex curves
Calc. Var. 7, 315–371 (1998) c Springer-Verlag 1998
Evolving convex curves Ben Andrews
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Mathematics Department, Stanford University, Stanford CA 94305, USA Received May 21, 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 xt : C → R2 of a fixed curve C , and the evolution equations considered have the following form: (1)
x˙ = −F n
where x˙ = ∂x ∂t , and F = F (κ, 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: 1 (2) F = ψ(n)κα , α 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 9 504 456 and a Terman fellowship. Mathematics Subject Classification: 35K15, 35K55, 35K65, 53A04 Present address: CMA, ANU, ACT 0200 AUSTRALIA
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flows (with F independent of n), and symmetric flows (with F (κ, n) = F (κ, −n) for all κ > 0 and all n ∈ S 1 ). Section I of the paper describes flows of the form (2) 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 I2.1). 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 C0 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 > 13 , 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 II2.8), 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 +2,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 II2.5). We expect that the subsequential convergence to the limiting shapes can be improved to C k +2,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 = 13 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 II1.12). The special case α = 13 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
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shape (Corollary II3.2). [A6] shows that the value 1 is optimal, by constructing examples of symmetric flows with any given α ∈ ( 13 , 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 [U1–2], Huisken [Hu4] and Gerhardt [Ge] have considered homogeneous equations with α = −1 in the isotropic case. Their results apply in the more complicated setting of hypersurfaces in higher dimensions, and the papers [U2] 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 [Ga1–2], [GH], and [Gr]. Some of the results on contracting anisotropic flows in the symmetric case appeared in [A1] (Section III, chapter 2), 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:
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∂ x =f ∂t
(0.1)
1 , n n. κ
In the homogeneous case we take β = −α, so that f (r, n) = β1 ψ(n)r β . Section I1 deals with the homogeneous case with smooth, strictly convex initial curves. Section I2 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 ) → R2 comprised of embeddings xt = x (., t) for each t, and satisfying the equation 1 ∂ x = ψ(n)κ−β n. ∂t β
(1.1)
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 Ct the curve xt (C ), and let At be the area enclosed by Ct . Theorem I1.1 Let x0 : C → R2 be a smooth, strictly convex embedding. Then there exists a unique solution x : C × [0, T ) → R2 of Eq. (1.1) with initial data x0 . The solution remains smooth and strictly convex on [0, T ). If 0 < β ≤ 1, then T = ∞ and q limt→∞ infp∈C |x (p, t)| = ∞. The rescaled curves given by the embeddings
π At
x (p, 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. If β > 1, then T < ∞ and as t approaches T , the inradius of Ct converges to infinity. The rescaled curves converge in the Hausdorff metric to a limit Σ. The 2, 1
support functions of the curves converge in C β 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 > 1 the result presented here is the best possible, in that there are examples of smooth positive ψ for which the support function of the homothetic 2, 1
solution is only C β . However there are several particular cases of interest where C ∞ convergence is attained: If ψ is constant, or has antipodal symmetry on S 1 , or has k -fold symmetry for some k ≥ 3; or if ψ is given explicitly in σ the form r[σ] β where σ is the support function of some smooth, strictly convex 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
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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 I1.2 below: The support function σ of Σ has continuous second derivatives, and the solution of Eq. (1.2) 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 C0 is defined by s(z ) = supx ∈C0 hx , z i for each z ∈ S 1 . Equivalently, if C0 is given by an embedding x : C → R2 then hx (p), n(p)i = s(n(p)) 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 R2 is smooth and strictly convex in nonradial directions is the support function of some smooth, strictly convex curve [A2, Eq. (2.10)]. The radius of curvature of the curve at the ∂2s point with normal z is given by r[s](z ) := ∂θ 2 (z ) + s(z ) where θ is the standard angle parameter on S 1 [A2, Eq. (2.11)]. Lemma I1.2 Suppose x : C × [0, T ) → R2 is a solution of Eq. (1.1). Then the support functions s : S 1 × [0, T ) → R satisfy the equation
(1.2)
1 ∂ s(z , t) = ψ(z ) ∂t β
∂2s (z , t) + s(z , t) ∂θ2
β .
Conversely, if s : S 1 × [0, T ) → R is a smooth solution of Eq. (1.2) with r[s] > 0, and if x0 : C → R2 is an smooth embedding with image o equal to the correspond∂s0 ∂z 1 ing initial curve C0 = s0 (z )z + ∂θ (z ) ∂θ : z ∈ S , then there exists a unique solution x : C × [0, T ) → R2 to Eq. (1.1) which has initial data x0 and such that the curve xt (C ) has support function st for each t ∈ [0, T ). Proof. For a smooth, strictly convex solution x of Eq. (1.1), we have s(z , t) = hx (n−1 (z )), z i where n−1 : S 1 → C is the inverse of the Gauss map n. The evolution of s can then be calculated as follows:
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∂ ∂ s(z , t) = hx (n−1 (z )), z i ∂t ∂t ∂ = h x (n−1 (z )), z i ∂t ∂ ∂ x (n−1 (z ) + Tx ( n−1 (z )), z i =h ∂t ∂t ∂ 1 = h ψ(z )κ(n−1 (z ))−β + Tx ( n−1 (z )), z i β ∂t 1 = ψ(z )r[s](z )β β ∂ −1 since Tx ( ∂t n (z )) is a vector tangential to Ct , and since r[s](z ) = κ(n−1 (z ))−1 as noted above. Conversely, if s : S 1 × [0, T ) → R is smooth, st = s(., t) extends to a convex homogeneous degree one function on R2 for each t, and s satisfies Eq. (1.2) then the embeddings x¯ : S 1 × [0, T ) → R2 defined by
(1.3)
x¯ (z , t) = s(z , t)z +
∂s ∂z (z , t) ∂θ ∂θ
define a set of curves {Ct } which have support function s. These embeddings satisfy an evolution equation which can be determined as follows: ∂ ∂ 2 s ∂z ∂ x¯ (z , t) = s(z , t)z + ∂t ∂t ∂t∂θ ∂θ 1 ∂ 1 ∂z = ψ(z )r[s](z )z + (ψ(z )r[s](z )) β β ∂θ ∂θ 1 = ψ (nx¯ (z )) (κx¯ (z ))−β + T x¯ (V ) β where nx¯ and κx¯ are the normal and curvature corresponding to the embedding x¯ , −β κx¯ ∂ ∂z 1 1 and V ∈ TS × [0, T ) is the vector field on S given by β ∂θ ψ(nx¯ )κx¯ ∂θ .
1 Here we have used the fact that T x¯ (V ) = κ−1 x¯ V for any V ∈ TS . The equation that x¯ satisfies is thus very similar to Eq. (1.1), differing only by the tangential term T x¯ (V ). We now proceed to construct a family of diffeomorphisms φ : C × [0, T ) → S 1 such that x (p, t) = x¯ (φ(p, t), t) gives a solution to Eq. (1.1). Take φ(p, 0) = nx0 (p) and solve the following ordinary differential equation for each p:
(1.4)
d φ(p, t) = −V (φ(p, t), t). dt
This equation has a unique solution for each p as long as s exists and remains smooth, and φt is a diffeomorphism for each t. The family of embeddings x satisfies x (p, 0) = x¯0 (nx0 (p)) = x0 (p), and evolves according to the following equation:
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∂ ∂ x (p, t) = x¯ (φ(p, t), t) ∂t ∂t ∂ ∂ = ( x¯ )(φ(p, t), t) + T x¯ ( φ(p, t), t) ∂t ∂t 1 = ψ(nx¯ (φ(p, t), t)) (κx¯ (φ(p, t), t))−β β + T x¯ (V (φ(p, t), t)) + T x¯ (−V (φ(p, t), t)) 1 = ψ(nx (p, t) (κx (p, t))−β β where we have used that nx (p, t) = nx¯ (φ(p, t), t) and κx (p, t) = κx¯ (φ(p, t), t). Hence x satisfies Eq. (1.1) as claimed. If xˆ is any other smooth solution of Eq. (1.1) giving rise to the same support functions s and agreeing with x0 at the initial time, then for xˆt and x¯x have the same image and hence are related by a diffeomorphism: xˆt = x¯ ◦ φˆ t . The above calculation in reverse shows that φˆ satisfies Eq. (1.4) with the same initial data as φ. By the uniqueness of solutions of the ordinary differential equation (1.4), φˆ ≡ φ and xˆ ≡ x . Lemma I1.2 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 I1.2, since Eq. (1.2) is uniformly parabolic: Given x0 , there exists a unique smooth solution for some time to Eq. (1.2) with initial condition s0 (z ) = hx0 (n−1 (z )), z i. Then the second part of Lemma I1.2 gives a solution x of Eq. (1.1) beginning at x . Suppose there are two solutions x and xˆ of Eq. (1.1). Then by the first half of Lemma I1.2 they give rise to solutions s and sˆ of Eq. (1.2) with the same initial condition, and since the equation is strictly parabolic we have s = sˆ . But the second half of Lemma I1.2 says that there is a unique solution of Eq. (1.1) corresponding to the solution s of Eq. (1.2), so x = xˆ . Lemma I1.4 If C0 has speed function f bounded below by a constant C1 > 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. (1.2) with respect to t: ∂ 1 ∂ ∂ β−1 s(z , t) = ψ(z )β (r[s](z , t)) s(z , t) r ∂t ∂t β ∂t (1.5) 2 ∂ ∂ β−1 ∂ β−1 s + ψr s . = ψr ∂θ2 ∂t ∂t This is a parabolic equation and the maximum principle applies to show that the ∂ s is nondecreasing. minimum of ∂t
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Lemma I1.5 For any smooth, strictly convex solution of Eq. (1.1) with support functions st , the following integral quantity is nonincreasing: 1
Z := A− 2
1 2π
Z S1
ψr[s]1+β d θ
1 1+β
,
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 6.2.1]) which states that for any two convex bodies K1 and K2 in the plane with corresponding support functions s1 and s2 , Z 2 Z Z s1 r[s1 ]d θ. s2 r[s2 ]d θ ≤ s1 r[s2 ]d θ . (1.6) S1
S1
S1
We note that this inequality still holds if one of the functions (say s1 ) has r strictly positive, and the other is an arbitrary smooth function, since the transformation s2 → s2 + Cs1 leaves the difference of the two sides of Eq. (1.6) unchanged, and a choice of sufficiently large C makes r[s2 ] positive, so that s2 becomes the support function of some convex region. Also, the equality case of Eq. (1.6) occurs precisely when s2 (z ) = cs1 (z ) + hz , ei for some constant c and some point e ∈ R2 . Now calculate the evolution equation satisfied by the integral quantity Z from Lemma I1.5: β (Z Z 2 ) − Z 1+β 1 ∂ β β β+1 Z= ψr d θ ψr r[ψr ] d θ − 1+β ∂t 2A S1 S1 βA 2 ≤0 since the bracket is just 1/A times the difference of the left and right hand sides ofREq. (1.6) in the case s1 = s and s2 = ψrβ : The enclosed area A is given by 1 2 S 1 sr[s] d θ. The equality case ocurs only when (1.7)
∂ s(z ) = cs(z ) − hz , ei ∂t
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for some e ∈ R2 and some positive c. The reader may verify directly that if Eq. (1.7) is satisfied at some time t0 , then the unique solution of Eq. (1.2) is given by (1.8) s(z , t) = hz , ei + ρ(t) s(z , t0 ) − hz , ei , 1
where ρ(t) = (1 + (1 − β)c(t − t0 )) 1−β for β = / 1 and ρ(t) = e c(t−t0 ) for β = 1. The curves Ct are the images of the embeddings x¯t given by Eq. (1.3), which are given by: x¯ (z , t) = e + ρ(t) (x¯ (z , t0 ) − e) .
(1.9)
Hence Eq. (1.7) is satisfied precisely when the solution curves evolve homothetically about the point e. 2
L Corollary I1.6 The isoperimetric ratio I = 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
Z\ = A− 2
(1.10)
Z S1
r[s]1+β d θ
1 1+β
≤
1 Z. infS 1 ψ
Now observe that a bound on Z\ implies a bound on the isoperimetric ratio, using the H¨older inequality: Z\ =
1 A1/2
Z
r[s]
1+β
S1
dθ
1 1+β
≥
1 β
(2π) 1+β A1/2
Z S1
β
− 1+β
r d θ = (2π)
L2 A
1/2 .
Since Z is bounded by its initial value, the result follows. 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 324). 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: 1−β 1−β ψrβ A 2 ψrβ A 2 decreases and inf Lemma I1.7 If 0 < β ≤ 1, sup s s increases. Proof. The evolution equation for A is computed as follows:
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Z 1 ∂ ∂ A= sr[s] d θ ∂t 2 ∂t S 1 Z 1 ψr1+β d θ. = β S1
(1.11)
It is useful to rewrite Eq. (1.2) in an alternative form: 1−β β ∂2 ∂ s = ψrβ−1 2 (s) + ψrβ−1 s + ψr . ∂t ∂θ β
(1.12)
Equations (1.11) and (1.12) combine with Eq. (1.5) to give an evolution equation 1−β
for
ψ(z )rβ A 2 s
∂ ∂t
: fA
1−β 2
s
! =
A
1−β 2
ψr
s
∂2 f + ψrβ−1 f ∂θ2
∂2s β−1 + ψr s + (1 − β)f ∂θ2 β+1 Z fA− 2 f r dθ + (1 − β) s S1 1−β ! 1−β ! 2 2ψrβ−1 ∂s ∂ fA 2 fA 2 β−1 ∂ + = ψr ∂θ2 s s ∂θ ∂θ s 1−β R f r dθ f fA 2 RS 1 − + (1 − β) s s sr d θ S1 1−β
−
(1.13)
β−1
fA 2 s2
ψrβ−1
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 2
vanishes. If 0 < β < 1, and a maximum of fA s occurs at a point z0 , we have f (z0 ) f (z0 ) f (z ) 1 s(z ) ≤ s(z0 ) for all z ∈ S , and therefore f (z ) ≤ s(z0 ) s(z ). Multiplying by r(z ) and integrating over z , we obtain R f (z0 ) 1 f r dθ ≥ RS , s(z0 ) sr d θ S1 and hence from Eq. (1.13) and the parabolic maximum principle (see for example 1−β 2
[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 supC ×[0,t0 ] |x | ≤ R, then
f (p, t0 ) ≤ max 2R sup t=0
f 2R − s
2R β supS 1 ψ , β(β − 1)β
.
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Proof. As before, we choose the origin to be enclosed by the initial curve, so that f obeys the following evolution equation, s is always positive. The quantity 2R−s which is a consequence of Eqs. (1.5) and (1.12): (1.14) f 2ψrβ−1 ∂s ∂ f ∂2 f ∂ = ψrβ−1 2 − ∂t 2R − s ∂θ 2R − s 2R − s ∂θ ∂θ 2R − s 2 R f β − 1 − − (2R − s)2 r 2 2ψrβ−1 ∂s ∂ f f ∂ − ≤ ψrβ−1 2 ∂θ 2R − s 2R − s ∂θ ∂θ 2R − s 1 − 1 ! β f2 supS 1 ψ β f − β − 1 − R (2R − s)2 Rβ 2R − s f where we have used that rβ ≥ supβ ψ β1 ψrβ = supβ ψ f ≥ supRβ ψ 2R−s S1
S1
since 2R − s is no less than R. Hence if the maximum of 2R β supS 1 ψ β(β−1)β
f 2R−s
then it must be decreasing. This gives a bound on bound on f since 2R − s ≤ 2R.
S1
is greater than
f 2R−s
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 as t → T . diam Ct → ∞ 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 (1.2) is uniformly parabolic on [0, T ), and consequently all higher derivatives of the support function remain bounded: C 2,α bounds follow from [K, Section 3.3] or from Lemma I1.12 below, and bounds on higher derivatives follow from Schauder theory. Hence the support functions s converge to a smooth limit sT as t approaches T . Since Eq. (1.2) 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. Proof. For β = 1, e
sup 1 ψ S t β
supt=0 s is a supersolution with bounded diameter for 1 1−β (1−β) supS 1 ψ 1−β + t . t < ∞. For 0 < β < 1 a supersolution is supt=0 s β The support function s of the solution is initially less than these supersolutions,
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and hence it remains so, and is bounded for all finite times. If β > 1 then − 1 (β−1) infS 1 ψ β−1 t is a subsolution which reaches infinite (inft=0 s)−(β−1) − β 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 ≤ 1 then Σ is smooth and strictly convex. If β > 1 then Σ is Lipschitz and has support function in C
2, β1
(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 I1.12 sup S 1 ×{t}
∂f ∂θ
Proof. ∂ ∂t
!
2 +f
∂f ∂θ
( ≤ max
2
!
2 +f
2
sup
S 1 ×{0}
∂f ∂θ
!
2 +f
2
) , sup f S 1 ×[0,t]
2
.
∂f ∂ ˙ f r[f ] + 2f f˙ r[f ] ∂θ ∂θ ∂f ˙ ∂ ∂2f 2 f r[f ] − 2 2 f˙ r[f ] = ∂θ ∂θ ∂θ + 2f f˙ r[f ]. =2
2 ∂f ∂ f + f = At a maximum of this quantity the derivative is zero, so we have 2 ∂θ 2 ∂θ 2 ∂f + f 2 > supt f 2 , then we also have 0. If the maximum is so large that ∂θ ∂f ∂θ
= / 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 I1.12 shows that the speed function satisfies a Lipschitz bound; equivalently, r[s]β is Lipschitz, so r[f ] is in C support function is in C
2, β1
0, β1
, and the
. Hence as before we have a subsequence which
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2, 1
converges to a C β function which has a zero time-derivative for the entropy, and hence is the support function of a homothetic solution of Eq. (1.1). 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 R 2 1 σ ds . Then solution of the evolution equation (1.1), let I = 4πA[C Ct t] I ≤1+
C . A[Ct ]
Consequently the ratio of the circumradius R(Ct , Σ) (the smallest factor by which Σ can be scaled while still enclosing a translate of Ct ) to the inradius r(Ct , Σ) (the largest factor by which Σ can be scaled while remaining enclosed by some translate of Ct ) converges to 1 as t approaches T . Proof. σ satisfies the following equation (see Eqs. (1.7–1.9)): (1.15)
ψ(z )r[σ]β = C βσ.
Hence ψ can be written in terms of σ, and the evolution equation (1.2) becomes β r[s] ∂ s = Cσ . (1.16) ∂t r[σ] Now we proceed to calculate the evolution R equation for the isoperimetric difference for the unrescaled solution: If V1 = S 1 σr[s]d θ, then (1.17) Z Z Z Z d 2 V − 4πA = 2 σr[s]d θ f r[σ]d θ − σr[σ]d θ f r[s]d θ dt 1 S1 S1 S1 S1 where we have used the formula (1.11) for theRenclosed area,R and the fact that Σ has enclosed area π, as well as the identity S 1 sr[σ]d θ = S 1 σr[s]d θ which follows by integrating by parts twice. Now we use the form of f and Eq. (1.15) together with the H¨older inequality: Z Z Z Z ψ β r[s] f r[s]d θ σr[σ]d θ = r[s]d θ σr[σ]d θ S1 S1 S1 β S1 β+1 Z Z r[s] C σr[σ] dθ σr[σ]d θ = r[σ] σ>0 σ>0 β Z Z r[s] r[s] dθ C σr[σ] dθ σr[σ] ≥ r[σ] r[σ] σ>0 Z Z σ>0 f r[σ]d θ σr[s]d θ = σ>0 S1 Z Z f r[σ]d θ σr[s]d θ. (1.18) = S1
S1
Thus the right-hand side of Eq. (1.17) is non-positive, and
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Z
2
S1
sr[σ]d θ
− 4πA ≤ C ,
where C is the initial value of the left hand side. Dividing by 4πA we deduce t ,Σ) the required estimate. The conclusion regarding the ratio R(C r(Ct ,Σ) follows from the Diskant inequality (see [Sc, Theorem 6.2.3]). Lemma I1.14 Let ς(z , t) = then
s(z ,t) σ(z ,t) .
If β = 1, then
sup ς(z ,t) inf ς(z ,t)
≤
(sup ς(z , 0))1−β + (1 − β)t (inf ς(z , 0))1−β + (1 − β)t
sup ς(z , t) ≤ inf ς(z , t)
sup ς(z ,0) inf ς(z ,0) .
!
If β ∈ (0, 1)
1 1−β
.
Proof. ς lies above the constant function inf ς(z , 0) initially, and below the constant function sup ς(z , 0), and evolves according to the equation !β β 2 r σ ∂2ς 1 2 ∂ς ∂ς ∂ς 1 = +ς = + ∂t β r[σ] β r[σ] ∂θ r[σ] ∂θ ∂θ By the maximum principle, ς is between inf ς(z , 0)1−β + (1 − β)t 1/(1−β) . sup ς(z , 0)1−β + (1 − β)t
1/(1−β)
and
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 − hpt , z i → 0 uniformly for some pt ∈ R2 . The speed bound of Lemma I1.8 and the derivative bound of Lemma I1.12 imply that the curvatures of the rescaled curves converge in C 1/β . It remains to show that pt converges to zero, which we prove in the following Lemmata.
Lemma I1.16 The quantity Q [s(t)] =
1+β
− A[s(t)] 2β
Z S1
ψ −1/β s
1+ β1
dθ
decreases, strictly unless s˜ = σ. 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 [s(t)] = Q [˜s (t)], and A[˜s (t)] = π. Since
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β−1 Z A 2 1 ψr[˜s ]β − ψr[˜s ]1+β d θ˜s , π 2π S 1 p we have (writing R(t) = A[Ct ]/π) Z 1+β d −1 ψ β s˜ β d θ dt S 1 Z 1+β ψ 1−1/β s˜ 1/β r[˜s ]β d θ = βR 1−β S1 Z Z 1 ψr[˜s ]1+β d θ ψ −1/β s˜ 1+1/β d θ − 2π S 1 S1 Z Z Z Z 1+β 1/β 1+1/β ξ d µ ξ d µ − ξ d µ d µ = 2πβR 1−β S1 S1 S1 S1 ≤0 ∂ s˜ = ∂t
by the H¨older inequality, where d µ = ψ −1/β s˜ 1+1/β d θ 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 pt 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 p(t) ≥ ε > 0. Assume without loss of generality that the initial curve q encloses the origin, so that s is always positive.
A[Ct ] . Define for each τ ∈ [0, T ) and each t ∈ As before write R(t) = π β−1 β−1 , R(τ ) (T − τ ) , −R(τ )
sˆ (θ, t, τ ) =
1 s θ, τ + R(τ )−(β−1) t . R(τ )
Then (1.19)
∂ sˆ = ψr[ˆs ]β . ∂t
The entropy estimate of Lemma I1.5 gives Z 1+β 2π 1 d ψr1+β d θ ≤ A= Z(0)1+β A 2 , dt β S1 β so that
π(β − 1) d 1−β Z(0)1+β . A 2 ≥− dt β
Integrating from τ to T and noting that limt→T A = ∞, we obtain
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(T − τ )R(τ )β−1 ≥ t0 =
β . (β − 1)(πZ(0))1+β
Therefore the solution sˆ is well-defined on S 1 × [0, t0 /2] × [0, T ). By the same argument, for each t the enclosed area of the curve with support function sˆ is bounded, independent of τ and t in this range. It follows (since the origin is enclosed) that the support function sˆ (θ, t, τ ) is bounded. The curvature bound of Lemma I1.8 and the speed gradient bound of Lemma I1.12 imply that sˆ (., t, τ ) is uniformly bounded in C 2,1/β . Since the speed is bounded, the Lipschitz constant of sˆ (θ, ., τ ) is bounded, independent of τ and θ. Note that sˆ (., 0, τ ) = s˜ (τ ). Therefore there is a subsequence of times τk approaching T , such that the functions sˆ (θ, t, τk ) on S 1 ×[0, t0 /2]×{τk } converge in C 2,γ in θ (for any γ < 1/β) and C 0,1 in t to a limiting function sˆ (θ, t, T ) which is C 2,1/β in θ and C 1 in t and satisfies Eq. (1.19). Furthermore, from Lemma I1.13 and the assumption that p(τ ) does not converge to zero we can assume that sˆ (θ, 0, T ) = σ(θ) + hp, z i, for some non-zero p ∈ R2 . By uniqueness of the solutions of Eq. (1.19), we have that sˆ (θ, t, T ) =
β−1 t 1− β
−
1 β−1
σ + hp, z i
for all t ∈ [0, t0 /2]. By Lemma I1.16, Q [ˆs (., t, T )] is strictly decreasing in t, so Q [ˆs (., 0, T )] − Q [ˆs (., t0 /2, T )] = ε > 0. Since Q [ˆs (., t, τk )] converges to Q [ˆs (., t, T )], we can choose k0 sufficiently large so that for all k ≥ k0 , we have 1 Q [ˆs (., 0, τk )] ≥ Q [ˆs (., 0, T )] − ε 3 and 1 Q [ˆs (., t0 /2, τk ] ≤ Q [ˆs (., t0 /2, T )] + ε. 3 Passing to a subsequence if necessary, assume that τk +1 > τk + R(τk )−(β−1) t0 /2 for all k ≥ k0 . Then since Q is decreasing, 0 ≥ Q [s(., τk +1 )] − Q [s(., τk + R(τk )−(β−1) t0 /2)] = Q [ˆs (., 0, τk +1 )] − Q [ˆs (., t0 /2, τk )] 1 1 ≥ Q [ˆs (., 0, T )] − ε − Q [ˆs (., t0 /2, T ) − ε 3 3 2 ≥ε− ε 3 1 ≥ ε, 3 a contradiction. This completes the proof of Theorem I1.1.
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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 > 1 there 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 for any γ > β1 .
2, β1
but not in C 2,γ
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 ∂2 −1 1 σ + σ = ψ βσβ. 2 ∂θ
(1.20)
In the case ψ = 1 this ordinary differential equation has solutions given by inverting the equation Z (1.21)
θ(σ) = θ0 + 0
σ
r
dr 1+β
2β β 1+β r
. −
r2
Now choose σ ≡ 0 on some interval [0, θ0 ] with θ0 < π. Continue on either side of this interval a short distance in S 1 by gluing in the solution from Eq. (1.21). This gives the support function of a convex curve which has a corner at the origin with angle θ0 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. (1.20) with ψ ≡ 1 on a region slightly larger σ 1 than [0, θ0 ] in S 1 , and ψ = r[σ] β which is smooth on S \[0, θ0 ]. The curve then evolves homothetically from the corner under Eq. (1.1). 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 infC |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. I2.
I2. 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:
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Theorem I2.1 Suppose C0 is a curve given by the boundary of an open convex region in R2 . Then there exists a unique family of convex curves Ct for 0 < t < T converging to C0 in the Hausdorff distance as t approaches zero, with support functions s(θ, t) satisfying Eq. (1.2). If 0 < β ≤ 1 then Ct 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 2,1/β support functions for positive times, and expand to infinite inradius, converging in C 2,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 C0 persists for some positive time. If Σ has bounded curvature then the curves Ct become strictly convex and smooth before the end of the interval of existence. 2, 1
Remark In the case β > 1 the support function is a classical (i.e C β ) solution of Eq. (I1.2) for positive times. The curves will in general be singular, however: The H¨older 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 I2.1 through the following series of a priori estimates: Lemma I2.2 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 diamC0 and supS 1 ψ: (2.1)
1
sup(s(z , t) − s(z , 0)) ≤ C (diam(C0 ), sup ψ)t 1+β . S1
S1
The speed of motion f is also bounded for positive times: (2.2)
sup f (z , t) ≤ C (diamC0 , sup ψ)t S1
β
− 1+β
.
S1
Proof (Lemma I2.2). We begin by proving the estimate (2.1). 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 (2.3)
1 ∂ x = (sup ψ)κ−β ∂t β S1
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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 = diamC0 is bounded, a sphere of any sufficiently large radius (at least d ) can be chosen to enclose C0 and to have √support function in direction z not exceeding that of C0 by more than R − R 2 − d 2 . Now let these spheres evolve by Eq. (2.3). The sphere at time 1 1−β sup ψt . If β > 1 this expression is only t has radius Rt = R 1−β + 1−β 1 S β valid for t
Evolving convex curves Ben Andrews
?
??
Mathematics Department, Stanford University, Stanford CA 94305, USA Received May 21, 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 xt : C → R2 of a fixed curve C , and the evolution equations considered have the following form: (1)
x˙ = −F n
where x˙ = ∂x ∂t , and F = F (κ, 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: 1 (2) F = ψ(n)κα , α 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 9 504 456 and a Terman fellowship. Mathematics Subject Classification: 35K15, 35K55, 35K65, 53A04 Present address: CMA, ANU, ACT 0200 AUSTRALIA
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flows (with F independent of n), and symmetric flows (with F (κ, n) = F (κ, −n) for all κ > 0 and all n ∈ S 1 ). Section I of the paper describes flows of the form (2) 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 I2.1). 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 C0 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 > 13 , 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 II2.8), 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 +2,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 II2.5). We expect that the subsequential convergence to the limiting shapes can be improved to C k +2,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 = 13 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 II1.12). The special case α = 13 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
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317
shape (Corollary II3.2). [A6] shows that the value 1 is optimal, by constructing examples of symmetric flows with any given α ∈ ( 13 , 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 [U1–2], Huisken [Hu4] and Gerhardt [Ge] have considered homogeneous equations with α = −1 in the isotropic case. Their results apply in the more complicated setting of hypersurfaces in higher dimensions, and the papers [U2] 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 [Ga1–2], [GH], and [Gr]. Some of the results on contracting anisotropic flows in the symmetric case appeared in [A1] (Section III, chapter 2), 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:
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B. Andrews
∂ x =f ∂t
(0.1)
1 , n n. κ
In the homogeneous case we take β = −α, so that f (r, n) = β1 ψ(n)r β . Section I1 deals with the homogeneous case with smooth, strictly convex initial curves. Section I2 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 ) → R2 comprised of embeddings xt = x (., t) for each t, and satisfying the equation 1 ∂ x = ψ(n)κ−β n. ∂t β
(1.1)
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 Ct the curve xt (C ), and let At be the area enclosed by Ct . Theorem I1.1 Let x0 : C → R2 be a smooth, strictly convex embedding. Then there exists a unique solution x : C × [0, T ) → R2 of Eq. (1.1) with initial data x0 . The solution remains smooth and strictly convex on [0, T ). If 0 < β ≤ 1, then T = ∞ and q limt→∞ infp∈C |x (p, t)| = ∞. The rescaled curves given by the embeddings
π At
x (p, 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. If β > 1, then T < ∞ and as t approaches T , the inradius of Ct converges to infinity. The rescaled curves converge in the Hausdorff metric to a limit Σ. The 2, 1
support functions of the curves converge in C β 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 > 1 the result presented here is the best possible, in that there are examples of smooth positive ψ for which the support function of the homothetic 2, 1
solution is only C β . However there are several particular cases of interest where C ∞ convergence is attained: If ψ is constant, or has antipodal symmetry on S 1 , or has k -fold symmetry for some k ≥ 3; or if ψ is given explicitly in σ the form r[σ] β where σ is the support function of some smooth, strictly convex 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
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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 I1.2 below: The support function σ of Σ has continuous second derivatives, and the solution of Eq. (1.2) 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 C0 is defined by s(z ) = supx ∈C0 hx , z i for each z ∈ S 1 . Equivalently, if C0 is given by an embedding x : C → R2 then hx (p), n(p)i = s(n(p)) 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 R2 is smooth and strictly convex in nonradial directions is the support function of some smooth, strictly convex curve [A2, Eq. (2.10)]. The radius of curvature of the curve at the ∂2s point with normal z is given by r[s](z ) := ∂θ 2 (z ) + s(z ) where θ is the standard angle parameter on S 1 [A2, Eq. (2.11)]. Lemma I1.2 Suppose x : C × [0, T ) → R2 is a solution of Eq. (1.1). Then the support functions s : S 1 × [0, T ) → R satisfy the equation
(1.2)
1 ∂ s(z , t) = ψ(z ) ∂t β
∂2s (z , t) + s(z , t) ∂θ2
β .
Conversely, if s : S 1 × [0, T ) → R is a smooth solution of Eq. (1.2) with r[s] > 0, and if x0 : C → R2 is an smooth embedding with image o equal to the correspond∂s0 ∂z 1 ing initial curve C0 = s0 (z )z + ∂θ (z ) ∂θ : z ∈ S , then there exists a unique solution x : C × [0, T ) → R2 to Eq. (1.1) which has initial data x0 and such that the curve xt (C ) has support function st for each t ∈ [0, T ). Proof. For a smooth, strictly convex solution x of Eq. (1.1), we have s(z , t) = hx (n−1 (z )), z i where n−1 : S 1 → C is the inverse of the Gauss map n. The evolution of s can then be calculated as follows:
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∂ ∂ s(z , t) = hx (n−1 (z )), z i ∂t ∂t ∂ = h x (n−1 (z )), z i ∂t ∂ ∂ x (n−1 (z ) + Tx ( n−1 (z )), z i =h ∂t ∂t ∂ 1 = h ψ(z )κ(n−1 (z ))−β + Tx ( n−1 (z )), z i β ∂t 1 = ψ(z )r[s](z )β β ∂ −1 since Tx ( ∂t n (z )) is a vector tangential to Ct , and since r[s](z ) = κ(n−1 (z ))−1 as noted above. Conversely, if s : S 1 × [0, T ) → R is smooth, st = s(., t) extends to a convex homogeneous degree one function on R2 for each t, and s satisfies Eq. (1.2) then the embeddings x¯ : S 1 × [0, T ) → R2 defined by
(1.3)
x¯ (z , t) = s(z , t)z +
∂s ∂z (z , t) ∂θ ∂θ
define a set of curves {Ct } which have support function s. These embeddings satisfy an evolution equation which can be determined as follows: ∂ ∂ 2 s ∂z ∂ x¯ (z , t) = s(z , t)z + ∂t ∂t ∂t∂θ ∂θ 1 ∂ 1 ∂z = ψ(z )r[s](z )z + (ψ(z )r[s](z )) β β ∂θ ∂θ 1 = ψ (nx¯ (z )) (κx¯ (z ))−β + T x¯ (V ) β where nx¯ and κx¯ are the normal and curvature corresponding to the embedding x¯ , −β κx¯ ∂ ∂z 1 1 and V ∈ TS × [0, T ) is the vector field on S given by β ∂θ ψ(nx¯ )κx¯ ∂θ .
1 Here we have used the fact that T x¯ (V ) = κ−1 x¯ V for any V ∈ TS . The equation that x¯ satisfies is thus very similar to Eq. (1.1), differing only by the tangential term T x¯ (V ). We now proceed to construct a family of diffeomorphisms φ : C × [0, T ) → S 1 such that x (p, t) = x¯ (φ(p, t), t) gives a solution to Eq. (1.1). Take φ(p, 0) = nx0 (p) and solve the following ordinary differential equation for each p:
(1.4)
d φ(p, t) = −V (φ(p, t), t). dt
This equation has a unique solution for each p as long as s exists and remains smooth, and φt is a diffeomorphism for each t. The family of embeddings x satisfies x (p, 0) = x¯0 (nx0 (p)) = x0 (p), and evolves according to the following equation:
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321
∂ ∂ x (p, t) = x¯ (φ(p, t), t) ∂t ∂t ∂ ∂ = ( x¯ )(φ(p, t), t) + T x¯ ( φ(p, t), t) ∂t ∂t 1 = ψ(nx¯ (φ(p, t), t)) (κx¯ (φ(p, t), t))−β β + T x¯ (V (φ(p, t), t)) + T x¯ (−V (φ(p, t), t)) 1 = ψ(nx (p, t) (κx (p, t))−β β where we have used that nx (p, t) = nx¯ (φ(p, t), t) and κx (p, t) = κx¯ (φ(p, t), t). Hence x satisfies Eq. (1.1) as claimed. If xˆ is any other smooth solution of Eq. (1.1) giving rise to the same support functions s and agreeing with x0 at the initial time, then for xˆt and x¯x have the same image and hence are related by a diffeomorphism: xˆt = x¯ ◦ φˆ t . The above calculation in reverse shows that φˆ satisfies Eq. (1.4) with the same initial data as φ. By the uniqueness of solutions of the ordinary differential equation (1.4), φˆ ≡ φ and xˆ ≡ x . Lemma I1.2 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 I1.2, since Eq. (1.2) is uniformly parabolic: Given x0 , there exists a unique smooth solution for some time to Eq. (1.2) with initial condition s0 (z ) = hx0 (n−1 (z )), z i. Then the second part of Lemma I1.2 gives a solution x of Eq. (1.1) beginning at x . Suppose there are two solutions x and xˆ of Eq. (1.1). Then by the first half of Lemma I1.2 they give rise to solutions s and sˆ of Eq. (1.2) with the same initial condition, and since the equation is strictly parabolic we have s = sˆ . But the second half of Lemma I1.2 says that there is a unique solution of Eq. (1.1) corresponding to the solution s of Eq. (1.2), so x = xˆ . Lemma I1.4 If C0 has speed function f bounded below by a constant C1 > 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. (1.2) with respect to t: ∂ 1 ∂ ∂ β−1 s(z , t) = ψ(z )β (r[s](z , t)) s(z , t) r ∂t ∂t β ∂t (1.5) 2 ∂ ∂ β−1 ∂ β−1 s + ψr s . = ψr ∂θ2 ∂t ∂t This is a parabolic equation and the maximum principle applies to show that the ∂ s is nondecreasing. minimum of ∂t
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Lemma I1.5 For any smooth, strictly convex solution of Eq. (1.1) with support functions st , the following integral quantity is nonincreasing: 1
Z := A− 2
1 2π
Z S1
ψr[s]1+β d θ
1 1+β
,
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 6.2.1]) which states that for any two convex bodies K1 and K2 in the plane with corresponding support functions s1 and s2 , Z 2 Z Z s1 r[s1 ]d θ. s2 r[s2 ]d θ ≤ s1 r[s2 ]d θ . (1.6) S1
S1
S1
We note that this inequality still holds if one of the functions (say s1 ) has r strictly positive, and the other is an arbitrary smooth function, since the transformation s2 → s2 + Cs1 leaves the difference of the two sides of Eq. (1.6) unchanged, and a choice of sufficiently large C makes r[s2 ] positive, so that s2 becomes the support function of some convex region. Also, the equality case of Eq. (1.6) occurs precisely when s2 (z ) = cs1 (z ) + hz , ei for some constant c and some point e ∈ R2 . Now calculate the evolution equation satisfied by the integral quantity Z from Lemma I1.5: β (Z Z 2 ) − Z 1+β 1 ∂ β β β+1 Z= ψr d θ ψr r[ψr ] d θ − 1+β ∂t 2A S1 S1 βA 2 ≤0 since the bracket is just 1/A times the difference of the left and right hand sides ofREq. (1.6) in the case s1 = s and s2 = ψrβ : The enclosed area A is given by 1 2 S 1 sr[s] d θ. The equality case ocurs only when (1.7)
∂ s(z ) = cs(z ) − hz , ei ∂t
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for some e ∈ R2 and some positive c. The reader may verify directly that if Eq. (1.7) is satisfied at some time t0 , then the unique solution of Eq. (1.2) is given by (1.8) s(z , t) = hz , ei + ρ(t) s(z , t0 ) − hz , ei , 1
where ρ(t) = (1 + (1 − β)c(t − t0 )) 1−β for β = / 1 and ρ(t) = e c(t−t0 ) for β = 1. The curves Ct are the images of the embeddings x¯t given by Eq. (1.3), which are given by: x¯ (z , t) = e + ρ(t) (x¯ (z , t0 ) − e) .
(1.9)
Hence Eq. (1.7) is satisfied precisely when the solution curves evolve homothetically about the point e. 2
L Corollary I1.6 The isoperimetric ratio I = 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
Z\ = A− 2
(1.10)
Z S1
r[s]1+β d θ
1 1+β
≤
1 Z. infS 1 ψ
Now observe that a bound on Z\ implies a bound on the isoperimetric ratio, using the H¨older inequality: Z\ =
1 A1/2
Z
r[s]
1+β
S1
dθ
1 1+β
≥
1 β
(2π) 1+β A1/2
Z S1
β
− 1+β
r d θ = (2π)
L2 A
1/2 .
Since Z is bounded by its initial value, the result follows. 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 324). 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: 1−β 1−β ψrβ A 2 ψrβ A 2 decreases and inf Lemma I1.7 If 0 < β ≤ 1, sup s s increases. Proof. The evolution equation for A is computed as follows:
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Z 1 ∂ ∂ A= sr[s] d θ ∂t 2 ∂t S 1 Z 1 ψr1+β d θ. = β S1
(1.11)
It is useful to rewrite Eq. (1.2) in an alternative form: 1−β β ∂2 ∂ s = ψrβ−1 2 (s) + ψrβ−1 s + ψr . ∂t ∂θ β
(1.12)
Equations (1.11) and (1.12) combine with Eq. (1.5) to give an evolution equation 1−β
for
ψ(z )rβ A 2 s
∂ ∂t
: fA
1−β 2
s
! =
A
1−β 2
ψr
s
∂2 f + ψrβ−1 f ∂θ2
∂2s β−1 + ψr s + (1 − β)f ∂θ2 β+1 Z fA− 2 f r dθ + (1 − β) s S1 1−β ! 1−β ! 2 2ψrβ−1 ∂s ∂ fA 2 fA 2 β−1 ∂ + = ψr ∂θ2 s s ∂θ ∂θ s 1−β R f r dθ f fA 2 RS 1 − + (1 − β) s s sr d θ S1 1−β
−
(1.13)
β−1
fA 2 s2
ψrβ−1
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 2
vanishes. If 0 < β < 1, and a maximum of fA s occurs at a point z0 , we have f (z0 ) f (z0 ) f (z ) 1 s(z ) ≤ s(z0 ) for all z ∈ S , and therefore f (z ) ≤ s(z0 ) s(z ). Multiplying by r(z ) and integrating over z , we obtain R f (z0 ) 1 f r dθ ≥ RS , s(z0 ) sr d θ S1 and hence from Eq. (1.13) and the parabolic maximum principle (see for example 1−β 2
[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 supC ×[0,t0 ] |x | ≤ R, then
f (p, t0 ) ≤ max 2R sup t=0
f 2R − s
2R β supS 1 ψ , β(β − 1)β
.
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325
Proof. As before, we choose the origin to be enclosed by the initial curve, so that f obeys the following evolution equation, s is always positive. The quantity 2R−s which is a consequence of Eqs. (1.5) and (1.12): (1.14) f 2ψrβ−1 ∂s ∂ f ∂2 f ∂ = ψrβ−1 2 − ∂t 2R − s ∂θ 2R − s 2R − s ∂θ ∂θ 2R − s 2 R f β − 1 − − (2R − s)2 r 2 2ψrβ−1 ∂s ∂ f f ∂ − ≤ ψrβ−1 2 ∂θ 2R − s 2R − s ∂θ ∂θ 2R − s 1 − 1 ! β f2 supS 1 ψ β f − β − 1 − R (2R − s)2 Rβ 2R − s f where we have used that rβ ≥ supβ ψ β1 ψrβ = supβ ψ f ≥ supRβ ψ 2R−s S1
S1
since 2R − s is no less than R. Hence if the maximum of 2R β supS 1 ψ β(β−1)β
f 2R−s
then it must be decreasing. This gives a bound on bound on f since 2R − s ≤ 2R.
S1
is greater than
f 2R−s
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 as t → T . diam Ct → ∞ 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 (1.2) is uniformly parabolic on [0, T ), and consequently all higher derivatives of the support function remain bounded: C 2,α bounds follow from [K, Section 3.3] or from Lemma I1.12 below, and bounds on higher derivatives follow from Schauder theory. Hence the support functions s converge to a smooth limit sT as t approaches T . Since Eq. (1.2) 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. Proof. For β = 1, e
sup 1 ψ S t β
supt=0 s is a supersolution with bounded diameter for 1 1−β (1−β) supS 1 ψ 1−β + t . t < ∞. For 0 < β < 1 a supersolution is supt=0 s β The support function s of the solution is initially less than these supersolutions,
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B. Andrews
and hence it remains so, and is bounded for all finite times. If β > 1 then − 1 (β−1) infS 1 ψ β−1 t is a subsolution which reaches infinite (inft=0 s)−(β−1) − β 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 ≤ 1 then Σ is smooth and strictly convex. If β > 1 then Σ is Lipschitz and has support function in C
2, β1
(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 I1.12 sup S 1 ×{t}
∂f ∂θ
Proof. ∂ ∂t
!
2 +f
∂f ∂θ
( ≤ max
2
!
2 +f
2
sup
S 1 ×{0}
∂f ∂θ
!
2 +f
2
) , sup f S 1 ×[0,t]
2
.
∂f ∂ ˙ f r[f ] + 2f f˙ r[f ] ∂θ ∂θ ∂f ˙ ∂ ∂2f 2 f r[f ] − 2 2 f˙ r[f ] = ∂θ ∂θ ∂θ + 2f f˙ r[f ]. =2
2 ∂f ∂ f + f = At a maximum of this quantity the derivative is zero, so we have 2 ∂θ 2 ∂θ 2 ∂f + f 2 > supt f 2 , then we also have 0. If the maximum is so large that ∂θ ∂f ∂θ
= / 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 I1.12 shows that the speed function satisfies a Lipschitz bound; equivalently, r[s]β is Lipschitz, so r[f ] is in C support function is in C
2, β1
0, β1
, and the
. Hence as before we have a subsequence which
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327
2, 1
converges to a C β function which has a zero time-derivative for the entropy, and hence is the support function of a homothetic solution of Eq. (1.1). 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 R 2 1 σ ds . Then solution of the evolution equation (1.1), let I = 4πA[C Ct t] I ≤1+
C . A[Ct ]
Consequently the ratio of the circumradius R(Ct , Σ) (the smallest factor by which Σ can be scaled while still enclosing a translate of Ct ) to the inradius r(Ct , Σ) (the largest factor by which Σ can be scaled while remaining enclosed by some translate of Ct ) converges to 1 as t approaches T . Proof. σ satisfies the following equation (see Eqs. (1.7–1.9)): (1.15)
ψ(z )r[σ]β = C βσ.
Hence ψ can be written in terms of σ, and the evolution equation (1.2) becomes β r[s] ∂ s = Cσ . (1.16) ∂t r[σ] Now we proceed to calculate the evolution R equation for the isoperimetric difference for the unrescaled solution: If V1 = S 1 σr[s]d θ, then (1.17) Z Z Z Z d 2 V − 4πA = 2 σr[s]d θ f r[σ]d θ − σr[σ]d θ f r[s]d θ dt 1 S1 S1 S1 S1 where we have used the formula (1.11) for theRenclosed area,R and the fact that Σ has enclosed area π, as well as the identity S 1 sr[σ]d θ = S 1 σr[s]d θ which follows by integrating by parts twice. Now we use the form of f and Eq. (1.15) together with the H¨older inequality: Z Z Z Z ψ β r[s] f r[s]d θ σr[σ]d θ = r[s]d θ σr[σ]d θ S1 S1 S1 β S1 β+1 Z Z r[s] C σr[σ] dθ σr[σ]d θ = r[σ] σ>0 σ>0 β Z Z r[s] r[s] dθ C σr[σ] dθ σr[σ] ≥ r[σ] r[σ] σ>0 Z Z σ>0 f r[σ]d θ σr[s]d θ = σ>0 S1 Z Z f r[σ]d θ σr[s]d θ. (1.18) = S1
S1
Thus the right-hand side of Eq. (1.17) is non-positive, and
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B. Andrews
Z
2
S1
sr[σ]d θ
− 4πA ≤ C ,
where C is the initial value of the left hand side. Dividing by 4πA we deduce t ,Σ) the required estimate. The conclusion regarding the ratio R(C r(Ct ,Σ) follows from the Diskant inequality (see [Sc, Theorem 6.2.3]). Lemma I1.14 Let ς(z , t) = then
s(z ,t) σ(z ,t) .
If β = 1, then
sup ς(z ,t) inf ς(z ,t)
≤
(sup ς(z , 0))1−β + (1 − β)t (inf ς(z , 0))1−β + (1 − β)t
sup ς(z , t) ≤ inf ς(z , t)
sup ς(z ,0) inf ς(z ,0) .
!
If β ∈ (0, 1)
1 1−β
.
Proof. ς lies above the constant function inf ς(z , 0) initially, and below the constant function sup ς(z , 0), and evolves according to the equation !β β 2 r σ ∂2ς 1 2 ∂ς ∂ς ∂ς 1 = +ς = + ∂t β r[σ] β r[σ] ∂θ r[σ] ∂θ ∂θ By the maximum principle, ς is between inf ς(z , 0)1−β + (1 − β)t 1/(1−β) . sup ς(z , 0)1−β + (1 − β)t
1/(1−β)
and
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 − hpt , z i → 0 uniformly for some pt ∈ R2 . The speed bound of Lemma I1.8 and the derivative bound of Lemma I1.12 imply that the curvatures of the rescaled curves converge in C 1/β . It remains to show that pt converges to zero, which we prove in the following Lemmata.
Lemma I1.16 The quantity Q [s(t)] =
1+β
− A[s(t)] 2β
Z S1
ψ −1/β s
1+ β1
dθ
decreases, strictly unless s˜ = σ. 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 [s(t)] = Q [˜s (t)], and A[˜s (t)] = π. Since
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329
β−1 Z A 2 1 ψr[˜s ]β − ψr[˜s ]1+β d θ˜s , π 2π S 1 p we have (writing R(t) = A[Ct ]/π) Z 1+β d −1 ψ β s˜ β d θ dt S 1 Z 1+β ψ 1−1/β s˜ 1/β r[˜s ]β d θ = βR 1−β S1 Z Z 1 ψr[˜s ]1+β d θ ψ −1/β s˜ 1+1/β d θ − 2π S 1 S1 Z Z Z Z 1+β 1/β 1+1/β ξ d µ ξ d µ − ξ d µ d µ = 2πβR 1−β S1 S1 S1 S1 ≤0 ∂ s˜ = ∂t
by the H¨older inequality, where d µ = ψ −1/β s˜ 1+1/β d θ 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 pt 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 p(t) ≥ ε > 0. Assume without loss of generality that the initial curve q encloses the origin, so that s is always positive.
A[Ct ] . Define for each τ ∈ [0, T ) and each t ∈ As before write R(t) = π β−1 β−1 , R(τ ) (T − τ ) , −R(τ )
sˆ (θ, t, τ ) =
1 s θ, τ + R(τ )−(β−1) t . R(τ )
Then (1.19)
∂ sˆ = ψr[ˆs ]β . ∂t
The entropy estimate of Lemma I1.5 gives Z 1+β 2π 1 d ψr1+β d θ ≤ A= Z(0)1+β A 2 , dt β S1 β so that
π(β − 1) d 1−β Z(0)1+β . A 2 ≥− dt β
Integrating from τ to T and noting that limt→T A = ∞, we obtain
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B. Andrews
(T − τ )R(τ )β−1 ≥ t0 =
β . (β − 1)(πZ(0))1+β
Therefore the solution sˆ is well-defined on S 1 × [0, t0 /2] × [0, T ). By the same argument, for each t the enclosed area of the curve with support function sˆ is bounded, independent of τ and t in this range. It follows (since the origin is enclosed) that the support function sˆ (θ, t, τ ) is bounded. The curvature bound of Lemma I1.8 and the speed gradient bound of Lemma I1.12 imply that sˆ (., t, τ ) is uniformly bounded in C 2,1/β . Since the speed is bounded, the Lipschitz constant of sˆ (θ, ., τ ) is bounded, independent of τ and θ. Note that sˆ (., 0, τ ) = s˜ (τ ). Therefore there is a subsequence of times τk approaching T , such that the functions sˆ (θ, t, τk ) on S 1 ×[0, t0 /2]×{τk } converge in C 2,γ in θ (for any γ < 1/β) and C 0,1 in t to a limiting function sˆ (θ, t, T ) which is C 2,1/β in θ and C 1 in t and satisfies Eq. (1.19). Furthermore, from Lemma I1.13 and the assumption that p(τ ) does not converge to zero we can assume that sˆ (θ, 0, T ) = σ(θ) + hp, z i, for some non-zero p ∈ R2 . By uniqueness of the solutions of Eq. (1.19), we have that sˆ (θ, t, T ) =
β−1 t 1− β
−
1 β−1
σ + hp, z i
for all t ∈ [0, t0 /2]. By Lemma I1.16, Q [ˆs (., t, T )] is strictly decreasing in t, so Q [ˆs (., 0, T )] − Q [ˆs (., t0 /2, T )] = ε > 0. Since Q [ˆs (., t, τk )] converges to Q [ˆs (., t, T )], we can choose k0 sufficiently large so that for all k ≥ k0 , we have 1 Q [ˆs (., 0, τk )] ≥ Q [ˆs (., 0, T )] − ε 3 and 1 Q [ˆs (., t0 /2, τk ] ≤ Q [ˆs (., t0 /2, T )] + ε. 3 Passing to a subsequence if necessary, assume that τk +1 > τk + R(τk )−(β−1) t0 /2 for all k ≥ k0 . Then since Q is decreasing, 0 ≥ Q [s(., τk +1 )] − Q [s(., τk + R(τk )−(β−1) t0 /2)] = Q [ˆs (., 0, τk +1 )] − Q [ˆs (., t0 /2, τk )] 1 1 ≥ Q [ˆs (., 0, T )] − ε − Q [ˆs (., t0 /2, T ) − ε 3 3 2 ≥ε− ε 3 1 ≥ ε, 3 a contradiction. This completes the proof of Theorem I1.1.
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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 > 1 there 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 for any γ > β1 .
2, β1
but not in C 2,γ
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 ∂2 −1 1 σ + σ = ψ βσβ. 2 ∂θ
(1.20)
In the case ψ = 1 this ordinary differential equation has solutions given by inverting the equation Z (1.21)
θ(σ) = θ0 + 0
σ
r
dr 1+β
2β β 1+β r
. −
r2
Now choose σ ≡ 0 on some interval [0, θ0 ] with θ0 < π. Continue on either side of this interval a short distance in S 1 by gluing in the solution from Eq. (1.21). This gives the support function of a convex curve which has a corner at the origin with angle θ0 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. (1.20) with ψ ≡ 1 on a region slightly larger σ 1 than [0, θ0 ] in S 1 , and ψ = r[σ] β which is smooth on S \[0, θ0 ]. The curve then evolves homothetically from the corner under Eq. (1.1). 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 infC |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. I2.
I2. 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:
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B. Andrews
Theorem I2.1 Suppose C0 is a curve given by the boundary of an open convex region in R2 . Then there exists a unique family of convex curves Ct for 0 < t < T converging to C0 in the Hausdorff distance as t approaches zero, with support functions s(θ, t) satisfying Eq. (1.2). If 0 < β ≤ 1 then Ct 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 2,1/β support functions for positive times, and expand to infinite inradius, converging in C 2,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 C0 persists for some positive time. If Σ has bounded curvature then the curves Ct become strictly convex and smooth before the end of the interval of existence. 2, 1
Remark In the case β > 1 the support function is a classical (i.e C β ) solution of Eq. (I1.2) for positive times. The curves will in general be singular, however: The H¨older 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 I2.1 through the following series of a priori estimates: Lemma I2.2 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 diamC0 and supS 1 ψ: (2.1)
1
sup(s(z , t) − s(z , 0)) ≤ C (diam(C0 ), sup ψ)t 1+β . S1
S1
The speed of motion f is also bounded for positive times: (2.2)
sup f (z , t) ≤ C (diamC0 , sup ψ)t S1
β
− 1+β
.
S1
Proof (Lemma I2.2). We begin by proving the estimate (2.1). 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 (2.3)
1 ∂ x = (sup ψ)κ−β ∂t β S1
Evolving convex curves
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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 = diamC0 is bounded, a sphere of any sufficiently large radius (at least d ) can be chosen to enclose C0 and to have √support function in direction z not exceeding that of C0 by more than R − R 2 − d 2 . Now let these spheres evolve by Eq. (2.3). The sphere at time 1 1−β sup ψt . If β > 1 this expression is only t has radius Rt = R 1−β + 1−β 1 S β valid for t
