The Affine Curve-Lengthening Flow $

J. reine angew. Math. 50 6 (1999), 43-83 Journal fur die reine und angewandte Mathematik © W a l t e r d e G r u y t e r Berlin · New York 1999

The affine curve-lengthening flow

B y Ben Andrews1) at Canberra

A b s t r a c t .The motion o f any smooth closed convex curve in the plane i n the d i r e c ¬ tion o f steepest increase o f its a f f i n e arc length can be continued smoothly f o r all time. The evolving curve remains strictly convex w h i l e expanding to i n f i n i t e s i z e and approaching a homothetically expanding ellipse.

1. Introduction

In this paper w e study an affine-geometric, fourth-order parabolic evolution equation f o r closed convex curves in the plane. This i s d e f i n e d by following the direction o f steepest ascent o f the a f f i n e arc length functional L, with respect to a natural affine-invariant inner product. The definitions o f these are g i v e n i n Section 2, along with a discussion o f other aspects o f the a f f i n e differential geometry o f curves.

The evolution equation can be written as follows:

( 1 ) $ where $ is the affine normal vector, and $ is the affine curvature. In terms of Euclidean- geometric invariants, this can be written as follows:

$ where  is the Euclidean curvature of the curve, u is an anticlockwise Euclidean arc-length parameter, n is the outward unit normal, and t is the Euclidean unit tangent vector. This is a fifth-order system of partial differential equations which is equivalent to a scalar fourth- order parabolic equation (see Section 3). Equation (1) is the parabolic analogue of the affine-geometric maximal surface equation, which has been an important problem in affine differential geometry, and with regard to which many problems remain unsolved (see [NS] and [LSZ]).

1) Research partly supported by NSF grant DMS 9504456 and a Terman fellowship.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 44 Andrews, The affine curve-lengthening flow

The analogous Euclidean-geometric evolution equation, the curve-shortening flow, has been studied extensively b y Gage ([Ga1], [Ga2]), Gage and Hamilton [GH], Grayson [Gr] and many others. For that evolution equation i t has been shown that any initial closed, smooth embedded curve produces a solution which exists f o r a f i n i t e time, contract- ing at the end o f that time to a single point, in such a w a y that the curves can be rescaled about the final point to converge to a circle. The curve-shortening flow i s a second-order partial differential equation, and many details of its analysis are based on the parabolic maximum principle, which i s not available to us i n the present problem. However, the result w e prove i n this paper i s somewhat analogous to that f o r the curve-shortening flow:

T h e o r e m 1 . Let $ be a smooth, strictly locally convex embedding of a curve $ into the plane. Then there exists a unique, smooth solution {x t } of th e ev o l u t i o n eq u a t i o n (1), which exists for all positive times. There exists a smooth embedding $, a point

$, and a real number $, such that the image of x∞ is an ellipse of enclosed area π with centre at the origin, and

Any e l l i p s e ev o l v e s b y expanding homothetically about its centre.

Some other evolution equations o f higher order f o r curves i n the plane have been considered before - Polden [Po1] has considered the L2 gradient flow o f the integral o f the square o f the curvature, which i s a fourth-order parabolic equation (note that Langer and Singer [LS] earlier introduced a d i f f e r e n t gradient flow, i n which the integral o f the squared curvature i s treated as a functional on the space o f W2,1 indicatrices o f curves o f fixed length - this produces a non-local flow o f zeroth order; W e n [Wl-2] has also considered the gradient flow o f the same functional on the space o f L2 indicatrices o f curves o f fixed length, which i s a second order parabolic equation). Higher order equations have also been considered f o r the evolution o f Riemannian metrics, b y Rendall [Re], Singleton [Si], Chruściel [Ch], and Polden [Po2].

The main resuit used in characterising the long-term behaviour of solutions (presented in Section 4) is closely related to integral estimates which the author developed in appli¬ cations to second-order evolution equations for curves and for hypersurfaces ([A1]-[A3]). The regularity estimates for the flow, given here in Section 7, are inspired by the work of Alexander Polden [Po1-2] on higher order evolution equations for curves and for metrics on surfaces. The equation under investigation here has some remarkable special features not shared by most other higher-order equations, however: First, we have the advantage of a large group of symmetries (namely, the special affine group); second, the equation has a certain degeneracy (see Eq. (19)) which has important consequences for the analysis - although in many situations such degeneracy may considered a disadvantage, it is vital to the argument of the present paper, because non-degenerate fourth-order equations do not preserve pointwise inequalities, and in particular could not keep arbitrary convex curves convex.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 45

The organisation o f the paper i s as follows: In Section 2 w e summarise various aspects o f the a f f i n e differential geometry o f curves, support functions and their use i n describing convex sets, and other results required i n the later analysis. In Section 3 w e introduce the evolution equation, establish the short-time existence o f solutions, and discuss the n o r m a ¬ lisation o f solution curves using scalings, translations, and area-preserving a f f i n e boosts. Section 4 accomplishes the k e y step i n the proof, using the Brunn-Minkowski theorem about concavity of the area functional to deduce an integral estimate with two important consequences: First, it implies that solutions which e x i s t f o r a l l time must evolve purely b y scaling i n the large-time limit; and secondly i t implies that the evolving curve remains strictly convex as long as it exists, as w e demonstrate i n Section 5 . Section 6 completes our characterisation o f the long-term behaviour b y showing that the only solutions which e v o l v e purely b y scaling are ellipses. This i s proved b y partially integrating the fourth-order ordinary differential equation characterising such solutions, and deriving a contradiction to the a f f i n e isoperimetric inequality i n a l l cases except ellipses. In particular w e make use o f an affine-geometric version o f the four-vertex theorem (Proposition 1 8 ) . Section 7 provides regularity estimates which imply that solutions e x i s t f o r a l l time, and Section 8 combines all these elements to complete the proof o f Theorem 1 .

W e note that the classification o f homothetic solutions f o r this evolution equation has also been carried out b y Lima and Montenegro [LM], and their analysis also g i v e s a classification o f non-embedded homothetic solutions. A d i f f e r e n t affine-geometric evolution equation, the f l o w b y the a f f i n e normal vector, has been studied b y Tannenbaum and Sapiro [TS1-2] and the author [A4]. That evolution equation i s o f second order, and the methods associated with it are more c l o s e l y related to those applied to the curve- shortening f l o w .

The author would like to thank Katsumi Nomizu f o r suggesting the equation considered here, and Alexander Polden f o r some f r u i t f u l discussions concerning the regularity o f solutions.

2. Background a n d notation

2.1. Support functions. Let $ be a closed, smooth, convex embedded curve in the plane, and Ω the region enclosed by $. Th e su p p o r t fu n c t i o n $ o f $ ( o r Ω) is d e f i n e d b y

( 2 ) $.

The support function contains all geometric information about the curve. In particular, there is a natural embedding $ with image equal to $, defined by

(3) $ for all $, where we think of S 1 as na t u r a l l y in c l u d e d in $, an d θ is the usual angle coordinate on S1. This embedding associates to a point z in S 1 the point of $ with normal direction z. The radius of curvature of $ at $ is given by

4 Journal für Mathematik. Band 506

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 46 Andrews, The affine curve-lengthening flow

( 4 ) r = hθθ + h, where the subscripts denote derivatives. More generally, for a C 2 fu n c t i o n $ w e d e f i n e r[f] = fθθ+f, and x[f]= f(z)z +fθzθ. These are related by the f a c t that $. Any function f with r [ f ] > 0 everywhere is the support function of a convex curve, given by the embedding $.

2.2. The Brunn-Minkowski Theorem. Let Ωi, i = 0,1 be bounded open convex sets in $, with support functions hi. Then their Minkowski sum Ω 0 + Ω l is th e co n v e x se t $. Let h be the support function of Ω 0 + Ω 1. Th e n

$, so support functions are additive under Minkowski addition.

The Brunn-Minkowski Theorem ([Br], [M], [Bl3], sections 21-22) concerns the concavity properties o f the area functional under Minkowski addition:

T h e o r e m 2 (The Brunn-Minkowski Theorem).For any bounded convex sets Ω0 and Ω1 in $, A [(1 — t)Ω0 + tΩ1]1/2 is a concave function of $, and is strictly concave unless c0Ω0 = c1Ω1 + p for some $ and some $.

The Brunn-Minkowski Theorem plays a crucial role i n the proof o f our main estimate i n Section 4.

2.3. A f f i n e d i f f e r e n t i a l g e o m e t r y o f c u r v e s .A f f i n e differential geometry concerns those properties o f $ which are invariant under the action o f area-preserving a f f i n e transforma¬ tions o f $. An affine-geometric arc-length element ds can be d e f i n e d b y

( 5 ) $ f o r any non-degenerate parameter u. S i n c e the quantity i n the bracket i s just the area o f the parallelogram formed b y dx/du and ∂2x/∂u2, and the special a f f i n e transformations preserve area and equivariantly transform the f i r s t and second derivatives, this definition i s affine-invariant; the invariance under reparametrisation i s clear. The a f f i n e arc-length parameter s is obtained by integrating ds. Then $ is the affine unit tangent vector, and $ is the affine normal vector.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 47

In order to write these in terms of Euclidean invariants, we take u to be Euclidean arc-length, t the Euclidean unit tangent vector x u, an d n the Euclidean unit normal. Then ds = 1 / 3 d u = r 2/3dθ where  is the Euclidean curvature of $ , so s is a no n - d e g e n e r a t e parameter if $ is strictly convex. In that case $ and

The definition of s , with the convention that the curve be parametrised anticlockwise, gives that $ everywhere, so $ and $ are linearly independent at each point, and we can define an affine-invariant inner product $ on the bundle $ over $ by taking $ and $ to be an orthonormal basis.

The affine curvature $ is defined by $. S i n c e $, w e h a v e

$. T h e r e f o r e $. I n terms of the support function, $.

I f V is an $- v a l u e d ve c t o r fi e l d on $, th e n $. D i f f e r e n - tiating, we obtain expressions for the derivatives of $ a n d $ a l o n g $ :

( 6 ) $ a n d

( 7 ) $.

W e w i l l a l s o find i t u s e f u l to c o n s i d e r the f u n c t i o n σ = hr1/3, w h i c h w e c a l l the affine support function. σ i s invariant under s p e c i a l l i n e a r transformations, but not translations.

Useful affine integral invariants are the enclosed area $ of the curve, the affine arc length $, and the total affine curvature $.

We note some useful identities: First, the enclosed area can be written as follows:

( 8 ) $.

T h e a f f i n e a r c l e n g t h c a n b e w r i t t e n i n s e v e r a l w a y s :

( 9 ) $ where we integrated by parts twice.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 48 Andrews, The affine curve-lengthening flow

Given $, write F = fr1/3. From the definition of x [f]we have

( 1 0 )

$. Differentiating, we find $, and the l e f t hand side can be rewritten as

Combining these, w e obtain the identity

( 1 1 ) $.

In particular, the cases F = σ and $ f o r $ yield x [f] = x an d x[f]= e respectively, and lead to the identities

( 1 2 ) $ and ( 1 3 ) $.

A f a m o u s r e s u l t i n a f f i n e g e o m e t r y i s the f o l l o w i n g , d ue to B l a s c h k e ( s e e [Bl1] or [Bl4], p.61):

T h e o r e m 3 (T h e af f i n e is o p e r i m e t r i c in e q u a l i t y ) . For any smooth, strictly convex curve $, with equality if and only if $ is an ellipse.

The following result i s also due to Blaschke i n the planar case [Bl2]:

T h e o r e m 4 (The Blaschke-Santaló inequality).For any smooth, strictly convex $ with support function h there exists a point $ enclosed by $ such that $

The equality can be made strict except when $ is an ellipse.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 49

2.4. Affine-geometric variation formulae. In this paper we consider theflowof steepest ascent of L wi t h re s p e c t to th e in n e r pr o d u c t $: That is, we wish to find a smooth family of embeddings xt starting from a prescribed embedding x0, such that

$ is as large as possible at each time $. In order to derive an expression for this evolution equation we consider the initial rate of change of L un d e r an ar b i t r a r y sm o o t h va r i a t i o n {xt}. For convenience we write $, and suppose without loss of generality that x0 is an affine-arc-length parametrisation of its image.

The variation of the arc length element ds can be calculated from (5): $

s i n c e a t t = 0 the curve is parametrised by affine arc length. The above definitions give $ and $, and we have $ everywhere. This gives

$ a n d

( 1 4 ) $

s i n c e du = ds at t = 0 . I n t e g r a t i o n o v e r t h e c u r v e g i v e s t h e v a r i a t i o n i n L:

( 1 5 )

The variation i n enclosed area can also be written i n affine-geometric language:

( 1 6 ) $.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 50 Andrews, The affine curve-lengthening flow

The total affine curvature changes as follows:

( 1 7 ) $

where w e used identity ( 1 1 ) i n the third line.

3. The evolution equation

It f o l l o w s f r o m Eq. ( 1 5 ) that the steepest ascent flow f o r the a f f i n e arc length with respect to the a f f i n e L2 metric i s (up to a factor) the evolution equation

This evolution equation i s equivalent to a scalar, fourth-order parabolic equation, s i n c e i n order to describe the evolving curve w e need only s p e c i f y the support function at each time. The support function e vol ve s according to the equation

( 1 8 ) $,

or more fully

(19) $

It can be seen that there are special solutions given by $ f o r a n y C, which correspond to expanding circles centred at the origin. Eq. ( 1 9 ) i s a nonlinear fourth- order scalar equation which i s parabolic provided hθθ + h > 0 everywhere. Proposition 2 . 3 o f [Lu] applies to show that i f the initial support function i s smooth and has r > 0 , then there exists a unique, smooth solution of equation ( 1 9 ) for a short time. This solution can then be used to reconstruct a unique solution o f the original equation ( 1 ) f o r a short time b y the same argument as i n [A1], Lemma I1.2.

The analysis of solutions of equation (1) is simplified by normalising the solution curves to have fixed enclosed area. Suppose $ is a solution of (1), and

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 51 d e f i n e $, where $. Then $ satisfies a modified evolution equation which w e derive as follows, where w e denote with a tilde those invariants associated with the rescaled solution $:

(20) $

since $

Since the equation (1) is equivariant under special affine transformations, it is also useful to normalise the solutions by performing translations to centre them as much as possible at the origin (in view of Theorem 4, it is useful to translate to minimise $), and also to apply optimal special linear transformations ( w e choose to minimise the Euclidean length$du o f the curve). These requirements are equivalent to the following: The enclosed area condition

(21a) $ the translation normalisation conditions

(21b) $ f o r all $, and the length-minimisation condition

(21c) $

f o r i = 1, 2 , wh e r e $ and $.In the remainder of the paper, normalised curves are smooth, strictly convex embedded closed curves satisfying the normalisation conditions (21a-c).

The resultant normalised solution, which we denote by $, sa t i s f i e s th e fo l l o w i n g evolution equation:

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 52 Andrews, The affine curve-lengthening flow

( 2 2 ) $

w h e r e $ c o s 2 θ an d $ s i n 2 θ; M is the matrix with elements

N is the matrix $

and finally, $.

Note that both M and N are positive definite, since tr $ and d e t $, w h i l e t r $ a n d

T h e s u p p o r t f u n c t i o n ĥ c o r r e s p o n d i n g t o t h e f u l l y n o r m a l i s e d c u r v e s $ e v o l v e s a c - c o r d i n g t o t h e e q u a t i o n

( 2 3 ) $

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 53

4. Convexity of the area functional

In this section w e prove a crucial estimate which w i l l control the (Euclidean) curvature o f the curve, and also imply that the curve i s asymptotically homothetically expanding. The resuit i s the following:

T h e o r e m 5 . For any smooth, strictly convex solution $ of Eq. (1), $ is a convex function of t, strictly convex unless there exists some c > 0 and some point $ such that $, in which case the curve is homothetically expand¬ ing about p: $

Proof.The statement o f this theorem has an obvious similarity to the statement o f the Brunn-Minkowski Theorem (Theorem 2), and indeed the main step i n the proof i s an application o f the Brunn-Minkowski Theorem.

In order to make u s e o f Theorem 2, w e f i r s t derive f r o m it a f a m i l y o f inequalities: Let $ be a f i x e d smooth, strictly convex curve enclosing a region Ω with support function h, and l e t f be a fixed smooth function on $. For ε s u f f i c i e n t l y small, the function h + εf i s the support function o f a smooth, strictly convex region Ω1. Consider the Brunn- Minkowski Theorem applied i n this case, with Ω0 = Ω. Then Ωt = ( 1 — t)Ω0 + tΩ1 has support function ht = ( 1 - t)h + t(h + εf) = h + tεf. Let F = fr[h]1/3 . W e compute:

$ a n d

where we used the identity (11). The concavity of A [Ωt]1/2 is equivalent to the inequality

Now integrate b y parts and note that F i s arbitrary to prove the following:

L e m m a 6 (Affine-geometric Wirtinger inequality). For any smooth, strictly convex embedded curve $ and any smooth $,

$, with equality if and only if $ for some $ and some $.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 54 Andrews, The affine curve-lengthening flow

Now we continue the proof of Theorem 5 by Computing the second time derivative o f A [ Ct]4/3, using the variation equations (16)—(17):

by Lemma 6 with $. Furthermore, by Lemma 6 equality occurs only if

for some constant c and some point $. Note that c cannot equal zero, since then we would have $. Multiplying by $ and integrating over $, we obtain using ( 7 ) w i t h V = q

and so q = 0 an d $, which is impossible in view of identity (9). Therefore, taking p = — q/c we find $. Hence the constant c must be positive, since other- wise we have $. S i n c e

w e h a v e

Thus i f x0 i s an affine-arc-length parametrisation o f $, Eq. ( 1 ) has the solution

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 55 w h e r e $, and φ(x, t) is obtained by solving for each ξ th e or d i n a r y d i f f e r e n t i a l equation

( 2 5 ) $, w h e r e $. □

The c o n v e x i t y o f the e n c l o s e d area has the f o l l o w i n g important consequence:

Corollary 7. For any smooth, strictly convex solution of Eq. (1), the quantity $

is non-negative and decreasing in time, strictly unless the solution curves are homothetically expanding about some centre.

The non-negativity o f $ f o l l o w s f r o m L e m m a 6 ( w i t h F = 1 ) together w i t h Theo- rem $ decreases since $ is non-negative by the proof of The- o r e m 5 , and s t r i c t l y positive u n l e s s the solution i s homothetic.

5. Preserving convexity a n d bounding the curvature

W e w i l l u s e the resuit o f Corollary 7 to d e d u c e upper and l o w e r bounds f o r the E u c l i d e a n curvature o f the solution $ o f the r e s c a l e d equation ( 2 2 ) . T o prepare f o r this, w e note the f o l l o w i n g automatic bounds w h i c h hold f o r a n y c o n v e x c u r v e w h i c h i s nor- m a l i s e d b y s p e c i a l a f f i n e transformations to s a t i s f y ( 2 1 a-c), and s o i n particular f o r the solutions o f Eq. ( 2 2 ) :

P r o p o s i t i o n 8 .There exist absolute positive constants c1 and c2 such that whenever $ satisfies ( 2 1 a-c),

for all $,where h is the support function of $.

Proof Write $. By (21c) we hâve for i = 1, 2

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 56 Andrews, The affine curve-lengthening flow

We compute

and similarly $. T h e r e f o r e

( 2 6 a ) $, a n d

( 2 6 b ) $.

I t f o l l o w s that h2 = 0 .

Then $ and, integrating against the positive function 1 ± cos k(θ —θk)f o r any$,w e obtain

$, so t h a t $ f o r e v e r y $.

Now w e u s e the f a c t that the enclosed area i s normalised to be π:

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 57

Therefore $. For all θ 0, we ha v e si n c e $

This proves the upper bound.

Fix θ0. Then $ is contained in $, s o h satisfies

Hence by Theorem 4,

S i n c e this approaches i n f i n i t y as h(θ0) approaches zero, h i s bounded below.□

Now w e proceed to our main estimate:

Theorem 9. For any normalised curve C, $

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 58 Andrews, The affine curve-lengthening flow

Proof. S i n c e $, w e h a v e

( 2 7 ) $.

B y the Hölder inequality, the Blaschke-Santaló inequality and the a f f i n e isoperimetric inequality w e have

( 2 8 ) $.

This gives the estimate

Define a length element dξ, by dξ = σ pds for p to be ch o s e n la t e r . Th e n we ha v e

Therefore

( 2 9 ) $.

The following resuit allows us to deduce that inf $ f o r a n y p:

L e m m a 1 0 . If $ is a smooth embedded convex curve with enclosed area π, then there exists $ with Σ(x0 ) = 1.

Proof. We have by Theorem 3, $, and so $. Thus the only possibility f o r the lemma to f a i l i s i f σ > 1 everywhere. Suppose this i s the case.

Let $ f o r a n y $. Then μ = 0 precisely at the two points of $ where $ is parallel to e. Let f = μ/σ, and dℓ = σds. Then we have $, and $ by (8), (13), and (12). Therefore

Also fss = σ2fℓℓ + σσℓfℓ, σs = σσℓ, and fs = σfℓ Combining these, we find $.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 59

I n t h e c a s e w h e r e σ > 1 e v e r y w h e r e , t h e S t u r m c o m p a r i s o n t h e o r e m ([Ha], T h e o r e m 3.1 of Chapter XI) implies that the interval of ℓ between zeroes of f is la r g e r th a n π . Since there are two zeroes of f , this implies that $, which is a contradiction, □

Now we choose two values of p in (2 9 ) to ge t th e tw o in e q u a l i t i e s in Th e o r e m 9: First choose p = 1, which gives $ a n d h e n c e

$, w h i c h i s t h e u p p e r b o u n d .

Next we take p = — 2, so th a t by Th e o r e m 4, $, a n d s o

$, w h i c h i s t h e l o w e r b o u n d i n T h e o r e m 9 . □

C o r o l l a r y 11.For any normalised curve $, we have

Proof. σ = hr1/3 and $ by Proposition 8, so the resuit follows directly from Theorem 9. □

This resuit shows that the normalised curves evolving under Eq. ( 2 2 ) remain uniformly convex and have uniformly bounded curvature. As another consequence o f this estimate w e have control above and below on the a f f i n e length o f the normalised curves:

Corollary 12. Any convex curve $ of enclosed area π satisfies $

Proof. We have $ by Theorem 9. □

6 . C l a s s i f i c a t i o n o f l i m i t s

In this section we show that the only smooth embedded convex curves which evolve by homothetic expansion are ellipses. Recall from the statement and proof of Theorem 5 that homothetically expanding solutions are those which satisfy the identity $ f o r some choice of origin and some c > 0. If $ sa t i s f i e s th i s id e n t i t y , th e n a su i t a b l y sc a l e d copy of $ satisfies $.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 60 Andrews, The affine curve-lengthening flow

T h e o r e m 1 3 .lf $ is a smooth, strictly convex embedded closed curve in $, satisfying the identity $ everywhere, then $ is an ellipse of enclosed area π centred at the origin.

Proof. The method o f proof i s to deduce a contradiction to the a f f i n e isoperimetric inequality i f $ i s not an ellipse. First, w e characterise ellipses:

L e m m a 1 4 . $ is an ellipse of enclosed area π centred at the origin if and only if σ ≡ 1.

Proof.I f $ i s an e l l i p s e centred at the origin with area π, then $ i s a special linear image o f the unit circle, and hence σ ≡ 1 b y the invariance o f σ.

I f σ ≡ 1 , then r = h-3. Therefore hθθ + h = h-3, and integration g i v e s

$ f o r some constant C. Also

Therefore $ for some $ and some θ0.

The support function of an ellipse may be calculated as follows: An ellipse of area π centred at 0 is the set $ for some symmetric matrix M with det M = 1. Therefore h(z) = $ is attained at a point with z = λMx for some constant λ, and taking the inner product with x gives h (z) = λ. Th u s x = h(z)-1M-lz, and the inner product with z gives h(z)2 = zTM-1 z for all $. Writing $

with ab — d2 = 1 , w e have

( 3 1 ) $

, where $. □

Next w e obtain a bound above on the enclosed area o f a homothetic solution:

L e m m a 1 5 . If $ then $, with equality if and only ifΣ≡1 .

Proof.S i n c e $, w e have

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 61 by Lemma 6 (with F = 1) together with Theorem 3. Hence $, and equality holds only if $ for some constant c and some $. But then

$ by (12) and (13). Therefore $, so t h a t c = 1 and p = 0. □

Finally w e consider the a f f i n e length L:

L e m m a 1 6 . If $, then either σ ≡ 1 or σ is periodic in s with period greater than $.

Proof.The a f f i n e support function σ can be computed explicitly in terms o f s as the solution of an ordinary differential equation: $, so we have

σss + σ 2 -1= 0 .

Hence $ for some constant E. Since σ is bo u n d e d we mu s t ha v e $, and we can write

where $, and $, and

which i s a decreasing function o f E. I f w e choose s = 0 to be at a point where σ attains its maximum, then ([AS], section 1 6 )

where $, which is decreasing in β an d he n c e increasing in E, an d sn ( u|m) is the Jacobian elliptic function defined by sn( u|m) = sin φ , $. Note that as E increases from — 4/3 to 4/3, m increases from 0 to 1.

It follows that the half-period S of σ is given in terms of the complete elliptic integral $

5 Journal für Mathematik. Band 506

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 62 Andrews, The affine curve-lengthening flow

L e m m a 1 7 . $, and S (m) is strictly increasing in m.

Proof. K(m) i s g i v e n f o r |m| < 1 by the convergent power s e r i e s ([AS], Equation 17.3.11) $

Hence w e also have $

Equality holds only f o r m = 0 .

I f w e write μ (m) = 1 — m + m2, then w e also have

f o r $, with equality if and only if m = 0. Combining these, we have

$ with equality only when m = 0. Hence $ is strictly increasing for m > 0, as required.

This completes the proof of Lemma 1 6 .□

It follows that $ for some integer $. Theorem 3 and Lemma 14 imply that $ if $ is not an ellipse, which is a contradiction except in the case k = 1. The following rules out that possibility:

P r o p o s i t i o n 1 8 (An a f f i n e four-vertex theorem).If $ is a closed, smooth, strictly convex embedded curve in $, then the affine curvature $ of $ has at least 4 critical points.

Proof. For any $, there is a unique point $ for which $. Define a function $ b y $. Then $ f o r a l l

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 63

$, so there exists some x0 such that ψ(x) = 0. Then $, and we denote this value b y$.

If there are less than 4 critical points of $, then there is a unique maximum and a unique minimum. Therefore $ consists of two components $ and $ such that $ f o r $, and $ f o r $.

D e f i n e $. Then $ only when x = x0 or $. Without loss of generality $ f o r $ and $ f o r $.

Note that $, and also $. Therefore $, and

But the construction above shows that the integrand is strictly positive on both $ and $. □

This completes the proof of Theorem 13, since if k = 1 th e n $ ha s on l y tw o cr i t i c a l points, contradicting Proposition 18, and if $ then $, contradicting The¬ orem 3 and Lemma 14. □

We note that Lima and Montenegro [LM] have also recently given a classification of the solutions of $, including non-embedded cases. Also, in the case where the curves are symmetric it can be shown that the affine isoperimetric ratio LA-1/3 is strictly increasing under Eq.(l) unless $ i s an ellipse, b y a method similar to that employed b y Gage [Ga1] f o r the curve shortening f l o w . However the symmetrisation argument used b y Gage to deduce this also f o r non-symmetric curves does not generalise easily to this situation because $ depends on third derivatives o f the curve, w h i l e the symmetrised curves are only C1,1.

7 . R e g u l a r i t y e s t i m a t e s

In this section w e g i v e the estimates required to prove that the evolving curves remain smooth, and that the solution exists f o r all time. These estimates depend strongly on the bound on $ obtained i n Section 4.

7.1. G a g l i a r d o - N i r e n b e r g e s t i m a t e s .First w e prove a slightly generalised version o f the Gagliardo-Nirenberg estimates, which w i l l be the main tool i n the proof o f the estimates o f this section:

Theorem 19. Given integers $ and $, and real numbers p, q, and r with $ and $, there exists a constant C depending on i, m, p, q, and r such that for any curve $ and any smooth function u on $ which attains a value of zero somewhere,

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 6 4 Andrews, The affine curve-lengthening flow

$ where

Proof. See [Au], page 93, for all cases except for i = 0, m = 1. Co n s i d e r th e la t t e r case, with p = ∞ : If q = 1 we ha v e $ trivially. If q > 1, de f i n e

$ f o r e a c h $. u is Hö l d e r co n t i n u o u s , with

If $ with|u ( x ) | = s u p | u|,then $ for a11 $. It followsthat

f o r a l l $, s i n c e u mu s t at t a i n ze r o so m e w h e r e . No w

A f t e r s o m e r e - a r r a n g e m e n t , t h i s b e c o m e s

$, where

$ a n d

I f i = 0 , m = 1 with any p > r, w e have b y the Hölder inequality

$ and the result follows by applying the previous case to ||u||∞.□

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 65

7.2. P r e l i m i n a r y e s t i m a t e s .W e derive some preliminary estimates which w i l l s i m p l i f y the calculations later:

Theorem 20. For any number $, there exist constants $ and $ such that every smooth embedded convex closed curve $ with enclosed area π and $ satisfies $.

Proof.S i n c e both sides o f the above equation are invariant under special a f f i n e transformations, w e can translate to s a t i s f y the condition (21b), and s o the estimates o f Propositions 8-12 hold. Also equations ( 2 7 ) and ( 2 8 ) show that

i n which both terms are non-negative. For convenience w e let u = — log σ. Then w e have

and $ by Eq. (12). Hence

( 3 2 ) $,

and ( 3 3 )

Now $

where we used the fact that u = 0 somewhere by Lemma 10, the bound on $ from Theorem 4, and the Poincaré inequality. Also $, and by Theorem 19, $.

Therefore $.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 66 Andrews, The affine curve-lengthening flow

A further application of Theorem 19 gives $

and, b y Theorem 9 and the Poincaré inequality,

Combining these g i v e s

In particular, for any C0, either $ or

Taking C0 sufficiently large (depending only on $), this implies if $ that

This implies by Theorem 19 that either $, or

proving the resuit. The application o f Theorem 1 9 to $ requires the following lemma:

L e m m a 2 1 . For any strictly convex normalised curve $ there exists a point $ such that $.

Proof. First note that

Hence the resuit holds unless $ everywhere. Let $ for any $ in $ at the two points of $ with $ parallel to e, and μ satisfies $ by (13). If $

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 67 everywhere, the interval o f s between zeroes i s greater than π b y the Sturm comparison theorem ([Ha], Chapter XI), s o L > 2π, contradicting Theorem 3.□

Corollary 22. For any $ and any $, every normalised curve $ wi t h $ satisfies

Proof.First w e prove that f o r any k,

( 3 4 ) $

This f o l l o w s f o r k = 1 b y applying the Cauchy-Schwarz inequality to the right-hand side o f the identity

( 3 5 ) $.

Proceeding by induction, suppose (34) holds for k = 1, ..., ℓ — 1, so in particular $

Applying the Cauchy-Schwarz inequality to

we obtain

$ Estimating the first factor on the left using (36), we find $ which g i v e s ( 3 4 ) f o r k = ℓ, completing the induction.

Now w e prove the corollary b y induction on k: Theorem 2 0 g i v e s the case k = 1 . Suppose the claim holds f o r k = ℓ — 1 :

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 68 Andrews, The affine curve-lengthening flow

Then if $, an application of (34) to the intégral on the right of the previous estimate g i v e s

$ which completes the induction after re-arrangement.□

C o r o l l a r y 23.For any k and $ there exists a constant C such that every normalised curve $ with $ satisfies $

Proof. By the identity (34), we have

Applying ( 3 4 ) again to the second factor on the right, w e find

Corollary 2 2 ( w i t h k replaced b y k + 2 ) applies to the first factor to g i v e the desired result. □

7.3. A bound on $.

Theorem 24. Let $ be a solution of Eq. (1). Then there exist con¬ stants C l, C2 and C3 depending only on $ such that

Proof. First we compute the evolution equation for $ under Eq. (1):

L e m m a 25. Under Eq. (1),

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 69

Proof. Recall that $, s o $. Therefore

Using the identity (11), we re-write this as

$. □

H e n c e t h e o n l y t e r m that m u s t b e c o n t r o l l e d i s t h e s e c o n d .

L e m m a 2 6 . For any ε > 0 and any $ there exists a constant $ such that every smooth, strictly convex embedded curve $ with $ satisfies

Proof. Note that his estimate is scaling-invariant: If $, then

$, and

Hence it suffices to consider the case where $.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 70 Andrews, The affine curve-lengthening flow

Now we estimate:

$, w h e r e w e u s e d the estimate ( 3 4 ) w i t h k = 1 , and the Cauchy-Schwartz inequality. The last term c a n b e estimated b y an application o f Theorem 1 9 , noting again that this c a n b e applied s i n c e $ i s zero s o m e w h e r e b y L e m m a 21:

Also, we estimate $ using Corollary 22. This gives

w h e r e the l a s t step f o l l o w s b y Y o u n g ’ s inequality.□

T o s i m p l i f y the proof o f Theorem 2 4 , w e note that there i s s o m e t i m e period o n w h i c h the e n c l o s e d area remains comparable to its initial value:

L e m m a 2 7 .For any solution of Eq. ( 1 ) ,

In particular, $ for $, where δ = 3/8 if $.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 71

Proof. This follows immediately from the identity $

together with the monotonicity of $ from Corollary 7. □

It follows from Lemmas 25 and 26 that we have for $

C o r o l l a r y 2 2 , t o g e t h e r w i t h t h e a r e a b o u n d , a p p l i e s t o t h e first t e r m t o g i v e

$, s o that

Hence, there exists $ such that either $ o r

T h e r e s u i t f o l l o w s b y i n t e g r a t i n g t h i s i n e q u a l i t y .□

S i n c e T h e o r e m 2 4 c a n b e a p p l i e d o v e r arbitrary t i m e i n t e r v a i s w i t h i n t h e i n t e r v a l o f e x i s t e n c e o f a s o l u t i o n , w e i m m e d i a t e l y o b t a i n t h e f o l l o w i n g :

Corollary 28. If $ is a smooth solution of Eq. (1), th e r e ex i s t s a co n s t a n t C depending only on $ and on $s u c h t h a t

$ for all $.

7 . 4 . H i g h e r r e g u l a r i t y .N e x t w e s e e k b o u n d s o n i n t e g r a l s o f d e r i v a t i v e s o f $. F i r s t w e c o m p u t e t h e r e q u i r e d e v o l u t i o n equations:

Proposition 29. Under Eq. (19), we can write $

where Pk is a polynomial of degree 3 + 2 k in $ and its derivatives up to order k + 2:

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 72 Andrews, The affine curve-lengthening flow

where $, and for each i

Proof.This f o l l o w s f o r k = 1 f r o m the proof o f Lemma 25, and f o r higher k b y induction using the identity

( 3 8 ) $

The resuit of Proposition 29, together with a computation of the time derivative of the length element ds under Eq. (1), gives an expression for the time derivative of the intégral of the square of any derivative of $. After integrating by parts all the terms which involve a single factor of $, we obtain the following:

Corollary 30. $

where β(i,j) is a non-negative integer, and for each i

As in the proof of the estimate on $, we will use the first term in this evolution equation to control a l l o f the remaining terms. This i s accomplished using the following result:

Proposition 31. Let k be a positive integer and0, β . . . , β k + 1 non-negative integers with $. Then for any ε > 0 and $ there exists a constant C

depending only on k, β0 , ..., βk + l, $ and ε, such that every smooth, strictly convex curve C with $ satisfies

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 73

Proof. As in Lemma 26, we can use the scaling invariance to reduce the problem to the case $. We first use the Hölder inequality, with an arbitrary non-trivial collection of exponents $: $

For each of the integrais on the right-hand side, we use Theorem 19. First, in the case j = 0, we have

$, where 2 (k + 2)γ0 = β0/2 — 1/p 0 . F o r j > 0, Theorem 19 gives

$ where 2 (k + 2)γj = (j + 1 / 2 ) β j —1/pj.In each o f these cases w e can apply Corollary 2 2 to bound the first factor, yielding: $

where $

Note that $

Therefore w e have

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 74 Andrews, The affine curve-lengthening flow

T h e resuit then f o l l o w s b y Y o u n g ’ s inequality, s i n c e

$ b e c a u s e $ a n d ∑ ( j + 2) βj = 2k + 8. □

W e are now i n a position to derive bounds on the integrais o f squares o f derivatives o f $ o f arbitrarily h i g h order:

Theorem 32. Let $ be a solution of Eq. (1). Th e n fo r an y in t e g e r $there exist constants C1, C2, and C 3 depending only on $ and k such that

$ for $, where

Proof. By Corollary 30 and Proposition 31, we have the estimate

Corollary 23 and scaling give

$ which we apply to the first term in (39). Lemma 27 implies that $ is comparable to $ for suitably small t . Writing $,

w e find that i f q i s s u f f i c i e n t l y large, then $,

which implies the resuit by comparison with the solution of the ordinary differential equation $. □

As in the case k = 0, th i s im p l i e s a sc a l i n g - i n v a r i a n t bo u n d as lo n g as th e so l u t i o n exists:

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 75

Corollary 33. Let $ be a smooth solution of Eq. (1), and write for each positive integer k $

Then there exists a constant C depending only on k, $ and qk(0), such that $ for all $.

8. The convergence theorem

In this section w e apply the results o f the previous sections to prove Theorem 1 . First w e show that solutions exist f o r a l l time:

T h e o r e m 34.Let x0 be a smooth, strictly convex embedded closed curve in $. Then the solution of Eq.(1) with initial conditions given by x0 exists for all positive times.

Proof.Suppose this i s not the case - that is, the solution exists only up to a f i n i t e time T. Note that the enclosed area $ i s bounded above and below on this time interval: Lemma 2 7 g i v e s a bound above f o r a i l time, and Theorem 3 g i v e s a bound below since the a f f i n e length i s increasing:

Therefore by Corollaries 28 and 33, $ and all of its derivatives are bounded on $. Under Eq. (19), we also have

so that the total change in the logarithm of r is bounded, and r remains bounded above and below. The evolution of the support function is given by

which is bounded, and so h re m a i n s bo u n d e d . It follows th a t th e cu r v e s $ ar e un i f o r m l y bounded in C k fo r ev e r y k. Fu r t h e r m o r e , th e pa r a m e t r i s a t i o n xt remain smooth and non-degenerate, since they are given by $,

( 4 0 ) $.

Hence the total change i n φ i s bounded, and b y differentiating ( 4 0 ) w e s e e that φ remains non-degenerate and the derivatives o f φ remain bounded.

It follows by the Arzela-Ascoli theorem that the maps xt converge to a limit x T for some subsequence of times approaching T, and xT is a smooth, non-degenerate parametri¬ sation of a strictly convex embedded closed curve. Since the time-derivative of x an d al l

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 76 Andrews, The affine curve-lengthening flow o f its derivatives remain bounded, xt converges i n C∞ to xTas t approaches T. The short-time existence theorem then shows that the solution can be extended beyond T, contradicting the maximality o f T.□

Next w e w i s h to show that the rescaled solutions converge to a limit as t approaches ∞ :

P r o p o s i t i o n 35.There exists a sequence of times { t k } approaching infinity, such that the curves $ given by the fully normalised e q u a t i o n ( 2 2 ) converge in C∞ to a unit circle centred at the origin.

Proof.W e note that the rescaled time parameter Τ also approaches i n f i n i t y as t does, s i n c e

$ and

$ s o

$ which approaches i n f i n i t y as t does.

Let ĥ(θ, Τ) be the support function o f the solution o f the f u l l y normalised equation (22). Then f o r each T > 0 d e f i n e ĥT(θ, Τ) = ĥT(θ, T + Τ). Then ĥT i s a solution o f the f u l l y normalised support function evolution equation (23).

Corollaries 2 8 and 3 3 g i v e uniform bounds on the a f f i n e curvature $ and its d e r i ¬ vatives f o r the f u l l y normalised solutions, f o r all time. B y Proposition 8 , the support function ĥ i s uniformly bounded above and below, and b y Corollary 11, $ i s uniformly bounded above and below. It f o l l o w s that the support function ĥ i s uniformly bounded i n C∞, with $ strictly positive.

Therefore there exists a sequence of times Tk such that $ converges i n C ∞ as k → ∞ to a limit $, which is therefore also a solution of (23).

W e claim that ĥ∞ ≡ 1 . B y Theorem 13, the equation $ i s s a t i s f i e d i f and only i f ĥ i s the support function o f an e l l i p s e o f enclosed area π centred at the origin, and s i n c e ĥ i s f u l l y normalised, this occurs i f and only if ĥ ≡1. B y Corollary 7 , the time derivative o f $ i s strictly négative i f this i s not the case. Choose a time interval [Τ0, Τ1] such that

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 77 f o r a l l $. Then choose k0 sufficiently large to ensure that

for all $ and all $. Choose a subsequence of {Tk}, also denoted {Tk}, such that for each k > k0, Tk + 1 + τ0> Tk+τ1.That is, all of the intervals [Tk + Τ0, Tk + Τ1] are disjoint for $.

Then

But this i s impossible s i n c e$ always.□

Stronger convergence to the limit w i l l f o l l o w f r o m a consideration o f behaviour f o r solutions close to ellipses: First w e show stability f o r the f u l l y normalised solutions.

Proposition 36. If ĥ: S1 × [0, ∞) satisfies (23), then for any k and any ε > 0 there exists C such that

P r o o f .F o r h c l o s e t o 1 , E q . ( 2 3 ) c a n b e w r i t t e n a s f o l l o w s :

( 4 1 )

$, where R is quadratic in h — 1 an d its derivatives up to fourth order. In order to control this, we note that many of these terms are negligible:

6 Journal für Mathematik. Band 506

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 78 Andrews, The affine curve-lengthening flow

L e m m a 37.For any normalised curve with support function h,

Proof. The first identity i s a restatement o f ( 2 1 c). The last i s a consequence o f ( 2 1 a), s i n c e $ s i n c e $.

The second identity comes from condition (21b), since (in view of the bounds above and below on h from Proposition 8), $. Integration against cosθ gives by (21b) $, and similarly with cos θ replaced b y s i n θ. □

This g i v e s the estimate

(42) $ where $ i s again quadratic i n h — 1 and its derivatives.

L e m m a 38.For any solution of Eq. ( 2 3 ) and any ε > 0 , t h e r e exists C(ε) such that

Proof.This f o l l o w s f r o m the form o f $, the uniform Ck bounds on the solution supplied b y Corollary 33, and Theorem 1 9 (applied with r = 2, and m s u f f i c i e n t l y large).□

Calculating the time derivative of $ using Eq. (42) and applying Lemma 3 8 w i t h ε = 1/4, we find

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 79

L e m m a 3 9 . If $ satisfies

$ ,

then

Proof. Expanding η in a trigonometric series $, we find

$, since $ is increasing for $. □

Hence the part o f h — 1 orthogonal i n L2 to 1 , cos(θ —θ0),and cos 2(θ —θ0)s a t i s f i e s this estimate, and b y Lemma 3 7 the remaining components can be bounded b y a∫(h — 1)2dθ and $. It f o l l o w s that f o r any ε > 0 there exists a radius r(ε) > 0 such that whenever $, $.

The result o f Proposition 3 6 follows, because the regularity estimates o f Corollary 3 3 allow any Wk,2 norm o f h — 1 to be bounded b y a power arbitrarily close to 1 o f the L2 norm ( b y taking m s u f f i c i e n t l y large i n Theorem 1 9 ) .□

Next w e use these stability estimates to control the scalings, special linear t r a n s f o r ¬ mations, and translations which relate the f u l l y normalised solutions to the un-normalised solutions of (19):

Proposition 40. Let h : S1 × [0, ∞) satisfy (19), and let ĥ be the corresponding solution of (23). Write $, where $, $ and $. There exists $, $ and t 0 and $ such that

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 80 Andrews, The affine curve- lengthening flow

$; $;

$; and

$ for any ε > 0 .

Proof.W e can recover the special linear transformations Tτ using the formula ( c o m ¬ pare Eq. ( 2 2 ) )

where $, and L1 and L2 are the Lie algebra elements given after Eq. (21c). Since the eigenvalues of N ar e bo u n d e d ab o v e an d be l o w , we ha v e

Now expanding about ĥ = 1 , w e f i n d

$, and b y Lemma 37, the first term on the right i s zero. Therefore b y Proposition 3 6 ,

Integrating, we deduce that $ f o r a n y ε > 0.

Next we consider the scalings R τ: We have $, and so $.

Note that $, so by Proposition 36 and Lem- ma 3 7 this i s bounded b y C(ε)e—(70/3—ε)τ. It f o l l o w s that

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 81

From the definition o f Τ i n terms o f t, this implies

Finally, w e consider the translations pt: W e have f r o m Eq. ( 2 3 )

where z1 = cosθ and z2 = sinθ, and

R4M has eigenvalues bounded above and below for large times, and so the square of the speed of translation is bounded by a constant times

Expanding about h = R(t) and using the bound from Lemma 37 on $, w e f i n d

The result f o l l o w s b y integrating this inequality.□

The final step in the proof o f Theorem 1 i s to show that the embedding xt g i v e n b y the solution of ( 1 ) also remain non-degenerate and converge, completing the proof o f Theorem 1 :

Proposition 41. If $ satisfies Eq. (1), and $ is the family of embeddings given by formula (3) ap p l i e d to th e su p p o r t fu n c t i o n h of th e corresponding solution of Eq. (19), then $ for some smooth family of embeddings $. There exists a smooth embedding $ such that φt converges in C ∞ to φ∞ as t approaches ∞.

P r o o f .B y computing the rate o f change o f the Euclidean unit normal under Eq. (1), w e find:

B y scaling and expanding about$,w e find:

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM 82 Andrews, The qffine curve-lengthening flow

The bounds on R(t) f r o m Proposition 4 0 and on ĥ— 1 f r o m Proposition 3 6 therefore g i v e

Hence the total change in φ i s bounded, and φ converges to a limiting map φ∞.

W e similarly bound the change in φθ, using the identity

Hence, estimating as above,

Therefore the limiting map φ∞ i s nondegenerate, hence an embedding. Similarly, the total change i n any higher derivative o f φ i s bounded, and so φ∞ i s a C∞ embedding and the convergence o f φt to φ∞ i s i n C∞.□

This completes the proof of Theorem 1, since the embeddings $ are smooth and non-degenerate whenever h is smooth and $.

References

[AS]M. Abramowitz and I. Stegun (Eds.), Handbook o f Mathematical Functions, Nat. Bur. Stand. Appl. Math. S e r . 5 5 ( 1 9 6 4 ) . [A1]B. Andrews, Evolving convex curves, Calc. V a r . and P.D.E., to appear. [A2]B. Andrews, Instability o f limiting shapes f o r curves evolving b y curvature, preprint. [A3]B. Andrews, Evolution o f hypersurfaces b y their Gauss curvature, preprint. [A4]B. Andrews, Contradiction o f convex hypersurfaces b y their a f f i n e normal, J. D i f f . Geom. 43 ( 1 9 9 6 ) , 2 0 7 — 2 3 0 . [Au]T. Aubin, Nonlinear analysis on manifolds, Springer-Verlag, New Y o r k 1 9 8 2 . [Bl1]W. Blaschke, Über a f f i n e Geometrie I: Isoperimetrische E i g e n s c h a f t e n von E l l i p s e und Ellipsoid, L e i p z i g e r B e r . 6 8 ( 1 9 1 6 ) , 2 1 7 — 2 3 7 . [Bl2]W. Blaschke, Über a f f i n e Geometrie VII: Neue Extremeigenschaften von E l l i p s e und Ellipsoid, L e i p z i g e r B e r . 69(1917), 3 0 6 — 3 1 8 . [Bl3]W. Blaschke, Kreis und Kugel, V e r l a g von V e i t & Comp., L e i p z i g 1 9 1 6 . [Bl4]W. Blaschke, V o r l e s u n g über Differentialgeometrie, V o l . II, Springer, B e r l i n 1 9 2 3 . [Br] H. Brunn, Ovale und E i f l ä c h e n , Thesis, München 1 8 8 7 . [Ch] P. T. Chruściel, S e m i - g l o b a l e x i s t e n c e and convergence o f solutions o f the Robinson-Trautman (2-dimen- sional Calabi) equation, Comm. Math. Phys. 1 3 7 ( 1 9 9 1 ) , 2 8 9 — 3 1 3 . [Ga1] M. Gage, An isoperimetric inequality with applications to curve shortening, Duke Math. J. 5 0 ( 1 9 8 3 ) , 1 2 2 5 — 1 2 2 9 . [Ga2] M. Gage, Curve shortening makes convex curves circular, Invent. Math. 7 6 ( 1 9 8 4 ) , 3 5 7 — 3 6 4 . [GH]M. Gage and R.S. Hamilton, The heat equation shrinking convex plane curves, J. D i f f . Geom. 2 3 ( 1 9 8 6 ) , 6 9 — 9 6 . [Gr] M. Grayson, The heat equation shrinks embedded plane curves to round points, J. D i f f . Geom. 2 6 (1987), 2 8 5 — 3 1 4 . [Ha] P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York 1964.

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Andrews, The affine curve-lenghtening flow 83

[LS] J. Langer and D. Singer, Curve straightening and a minimax argument f o r c l o s e d elastic curves, Topology 2 4 ( 1 9 8 5 ) , 7 5 — 8 8 . [LSZ] A.M. Li, U. Simon, and G.S. Zhao, Global a f f i n e d i f f e r e n t i a l geometry o f hypersurfaces, de Gruyter Expos. Math. 1 1 , W a l t e r d e Gruyter & Co., B e r l i n 1 9 9 3 . [LM] L. Lima and F. Montenegro, Classification o f homothetic solutions f o r the a f f i n e curvature f l o w , preprint. [Lu] A. Lunardi, An introduction to geometric theory o f f u l l y nonlinear parabolic equations, in: Qualitative aspects and applications o f nonlinear evolution equations, T.T.Li and P . de Mottoni (Eds.), W o r l d S c i e n t i f i c , Singapore 1991. [M] H. Minkowski, V o l u m e n und Oberfläche, Math. Ann. 5 7 ( 1 9 0 3 ) , 4 4 7 — 4 9 5 . [NS] H. Nomizu and T. Sasaki, A f f i n e d i f f e r e n t i a l geometry: geometry o f a f f i n e immersions, Cambridge U n i ¬ versity Press, 1 9 9 4 . [Po1] A. Polden, Closed curves o f l e a s t total curvature, preprint, Tübingen, 1 4 pages. [Po2] A. Polden, Compact surfaces o f least total curvature, preprint, Tübingen, 1 6 pages. [Re] A. D. Rendall, Existence and asymptotic properties o f global solutions o f the Robinson-Trautmann e q u a ¬ tion, Class. Quant. Grav. 5 ( 1 9 8 8 ) , 1 3 3 9 — 1 3 4 7 . [ST1] G. Sapiro and A. Tannenbaum, O n invariant curve evolution and image analysis, Indiana Univ. J. Math. 42 ( 1 9 9 3 ) , 9 8 5 — 1 0 0 9 . [ST2] G. Sapiro, O n a f f i n e plane curve evolution, J. Funct. Anal. 1 1 9 ( 1 9 9 4 ) , 7 9 — 1 2 0 . [Sc] R. Schneider, Convex bodies: The Brunn-Minkowski theory, Encyl. Math. Appl. 44, Cambridge University Press, 1 9 9 3 . [Si] D. Singleton, On global existence and convergence o f vacuum Robinson-Trautman solutions, Class. Quant. Grav. 7 ( 1 9 9 0 ) , 1 3 3 3 — 1 3 4 3 . [W1] Y. Wen, L2 f l o w o f curve straightening i n the plane, Duke Math. J. 70 ( 1 9 9 3 ) , 6 8 3 — 6 9 8 . [W2] Y. Wen, Curve straightening f l o w deforms c l o s e d plane curves w i t h non-zero rotation number to circles, J. D i f f . Equ. 1 2 0 ( 1 9 9 5 ) , 8 9 — 1 0 7 .

Department o f Mathematics, Australian National University, Canberra, ACT 0 2 0 0 , Australia

Eingegangen 14. November 1997

Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM Brought to you by | Library (Chifley) BLG 15 Authenticated Download Date | 9/22/15 3:49 AM