The curve shortening problem [1 ed.] 1584882131, 9781584882138, 9781420035704

Citation preview

Curve Shortening Problem The

© 2001 by Chapman & Hall/CRC

Curve Shortening Problem The

Kai-Seng Chou Xi-Ping Zhu

CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C.

© 2001 by Chapman & Hall/CRC

Library of Congress Cataloging-in-Publication Data Chou, Kai Seng. The curve shortening problem / Kai-Seng Chou, Xi-Ping Zhu. p. cm. Includes bibliographical references and index. ISBN 1-58488-213-1 (alk. paper) 1. Curves on surfaces. 2. Flows (Differentiable dynamical systems) 3. Hamiltonian sytems. I. Zhu, Xi-Ping. II. Title. QA643 .C48 2000 516.3′52—dc21

00-048547

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

© 2001 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-5848-213-1 Library of Congress Card Number 00-048547 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

© 2001 by Chapman & Hall/CRC

v CONTENTS

Prefa e 1

vii

Basi Results

1

1.1 Short time existen e . . . . . . . . . . . . . . . . . . . 1 1.2 Fa ts from the paraboli theory . . . . . . . . . . . . . 15 1.3 The evolution of geometri quantities . . . . . . . . . . 19 2

Invariant Solutions for the Curve Shortening Flow

2.1 2.2 2.3 2.4 3

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Blas hke Sele tion Theorem . . . . . . . . . . Preserving onvexity and shrinking to a point Gage-Hamilton Theorem . . . . . . . . . . . . The ontra ting ase of the ACEF . . . . . . The stationary ase of the ACEF . . . . . . . The expanding ase of the ACEF . . . . . . .

. . . . . .

. . . . . .

Results from the Brunn-Minkowski Theory The AGCSF for  in (1/3, 1) . . . . . . . . The aÆne urve shortening ow . . . . . . . Uniqueness of self-similar solutions . . . . .

The Non- onvex Curve Shortening Flow

. . . .

. . . .

. . . .

27 29 33 35 45

. . . . . .

. . . . . .

. . . . . .

The Convex Generalized Curve Shortening Flow

4.1 4.2 4.3 4.4 5

. . . .

The Curvature-Eikonal Flow for Convex Curves

3.1 3.2 3.3 3.4 3.5 3.6 4

Travelling waves . . . . . . . . . . . . . Spirals . . . . . . . . . . . . . . . . . . . The support fun tion of a onvex urve Self-similar solutions . . . . . . . . . . .

27

. . . .

45 47 51 59 73 80 93

. . . .

. . . .

94 97 102 112 121

5.1 An isoperimetri ratio . . . . . . . . . . . . . . . . . . 121 5.2 Limits of the res aled ow . . . . . . . . . . . . . . . . 129

© 2001 by Chapman & Hall/CRC

vi

5.3 Classi ation of singularities . . . . . . . . . . . . . . . 134 6

A Class of Non- onvex Anisotropi Flows

6.1 6.2 6.3 6.4 6.5 7

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Embedded Closed Geodesi s on Surfa es

7.1 7.2 7.3 7.4 8

The de rease in total absolute urvature The existen e of a limit urve . . . . . . Shrinking to a point . . . . . . . . . . . A whisker lemma . . . . . . . . . . . . . The onvexity theorem . . . . . . . . . .

143

Basi results . . . . . . . . The limit urve . . . . . . Shrinking to a point . . . Convergen e to a geodesi

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

144 147 153 160 164 179

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

180 186 188 196

The Non- onvex Generalized Curve Shortening Flow

8.1 8.2 8.3 8.4 8.5

Short time existen e . . . . . . . The number of onvex ar s . . . The limit urve . . . . . . . . . . Removal of interior singularities . The almost onvexity theorem . .

Bibliography

© 2001 by Chapman & Hall/CRC

203

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

204 211 218 228 239 247

vii

PREFACE

A geometri evolution equation (for plane urves) is of the form  = fn ; t

()

where (
; t) is a family of urves with a hoi e of ontinuous unit normal ve tor n(
; t) and f is a fun tion depending on the urvature of (
; t) with respe t to n(
; t). Any solution of () is invariant under the Eu lidean motion. The simplest geometri evolution equation is the eikonal equation when f is taken to be a non-zero onstant. The next one is the urvature-eikonal ow when f is linear in the

urvature. It in ludes the urve shortening ow (CSF)  = kn ; t

as a spe ial ase. Let L(t) be the perimeter of a family of losed

urves (
; t) driven by (). We have the rst variation formula Z dL f kds : (t ) = dt

(
;t)

Therefore, the CSF is the negative L2 -gradient ow of the length. When (
; t) is also embedded, its en losed area satis es dA (t) = dt

2 :

Thus, any embedded losed urve shrinks under the ow and eases to exist beyond A(0)=2. The following two results ompletely hara terize the motion. The CSF preserves onvexity and shrinks any losed onvex urve to a point. Furthermore, if we dilate the ow so that its en losed area is always equal to , the normalized

ow onverges to a unit ir le.

Theorem A (Gage-Hamilton)

The CSF starting at any losed embedded

urve be omes onvex at some time before A(0)=2.

Theorem B (Grayson)

© 2001 by Chapman & Hall/CRC

viii

From the analyti point of view, the urvature of ( ; t) satis es kt = kss + k 3 ; where s = s(t) is the ar -length parameter of ( ; t). This is a nonlinear heat equation with superlinear growth. It is lear that the

urvature must blow up in nite time. However, it is the geometri nature of the ow that enables one to obtain pre ise results like these two theorems. On the other hand, the CSF is a spe ial ase of the mean urvature ow for hypersurfa es. It turns out that, although Theorem A ontinues to hold for the mean urvature ow, Theorem B does not. This makes planar ows spe ial among urvature ows. After Theorems A and B, subsequent works on the CSF go in two dire tions. One is to study the stru ture of the singularities of the ow for immersed urves, and the other is to onsider more general planar ows. In this book, we present a omplete treatment on Theorem A and Theorem B as well as provide some of generalizations. There are eight hapters. We outline the ontent of ea h

hapter as follows: In Chapter 1, we dis uss basi results su h as lo al existen e, separation prin iple, and niteness of nodes for the general ow ( ) under the paraboli assumption. In Chapter 2, we des ribe spe ial solutions of the CSF whi h arise from its Eu lidean and s aling invarian e: travelling waves, spirals, and ontra ting and expanding self-similar solutions. These solutions will be ome important in the lassi ation of singularities for the CSF. Theorem A is proved in Chapter 3. In the same hapter, we also study the anisotropi urvature-eikonal ow







 t

= ((n)k + (n)) ;  > 0 : This ow may be viewed as the CSF in a Minkowski geometry when 0 and its general form is proposed as a model in phase transition. Depending on the inhomogeneous term , it shrinks to a point, expands to in nity, or onverges to a stationary solution. We determine its asymptoti behaviour in all these ases. In Chapter 4, we study the anisotropi generalized CSF 

 t

= (n) k  j

j

1

kn ;

>0;

>0;

following the work of Andrews [8℄ and [10℄. Analogues of Theorem A are proved for  [1=3; 1). When  = 1=3 and  1, the ow is 2

© 2001 by Chapman & Hall/CRC



ix

aÆne invariant and is proposed in onne tion with image pro essing and omputer vision. Beginning from Chapter 5, we turn to non onvex urves. First, we present a relatively short proof of Theorem B whi h is based on the blow-up and the lassi ation of singularities. This approa h has been su

essfully adopted in many geometri problems in luding nonlinear heat equations, harmoni heat ows, Ri

i ows, and the mean urvature ow. Next, we present Grayson's geometri approa h where the Sturm os illation theorem is used in an essential way in Chapter 6. Though stri tly two-dimensional, it is powerful and works for a large lass of uniformly paraboli ows (). In Chapter 7, we dis uss how the CSF an be used to prove the existen e of embedded, losed geodesi s on a surfa e. Finally, in Chapter 8, we study the isotropi generalized CSF and establish an almost onvexity theorem when  2 (0; 1). Whether the onvexity theorem holds for this lass of ows remains an unsolved problem. Many interesting results on () have been obtained in the past fteen years. It is impossible to in lude all of them in a book of this size. Apart from a thorough dis ussion on Theorem A and B, the

hoi e of the rest of the material in this book is rather subje tive. Some are based on our work on this topi . To balan e things the we sket h the physi al ba kground, des ribe related results, and o

asionally point out some unsolved problems in the notes whi h an be found at the end of ea h hapter. We hope that the reader an gain a panorami view through them. We shall not dis uss the levelset approa h to urvature ows in spite of its popularity. Here we are mainly on erned with singularities and asymptoti behaviour of planar ows where the lassi al approa h is suÆ ient. Thanks are due to Dr. Sunil Nair for proposing the proje t, and to Ms.Judith Kamin for her eort in editing the book. We are also indebted to the Earmarked Grant of Resear h, Hong Kong, the Foundation of Outstanding Young S holars, and the National S ien e Foundation of China for their support in our work on urvature ows, some of whi h has been in orporated in this book.

© 2001 by Chapman & Hall/CRC

Chapter 1

Basi Results In this hapter, we rst establish the existen e of a maximal solution and some basi qualitative behaviour su h as the separation prin iple and niteness of nodes for the general ow (1.2). These properties are dire t onsequen es of the paraboli nature of the ow. For the reader's onvenien e, we olle t fundamental results on paraboli equations in Se tion 2. In parti ular, the \Sturm os illation theorem," whi h is not found in standard texts on this subje t, will play an important role in the removal of singularities of the ow. In Se tion 3, we derive evolution equations for various geometri quantities of the ow. They will be ome important when we study the long time behaviour of the ow.

1.1

Short time existen e

We begin by re alling the de nition of a urve. An immersed, C 1 urve is a ontinuously dierentiable map from I to R2 with a non-zero tangent p = d =dp. Throughout this book, I is either an interval or an ar of the unit ir le S 1 , and a urve always means an immersed, C 1 - urve unless spe i ed otherwise. The urve is losed 1 © 2001 by Chapman & Hall/CRC

2

CH. 1.

Basi Results

if I is the unit ir le. It is embedded if it is one-to-one. Given a

urve = ( 1 ; 2 ), its unit tangent is given by t = p =j p j and its unit normal, n, is given by ( p2 ; p1 )=j p j. When is an embedded

losed urve and t runs in the ounter lo kwise dire tion, n is the inner unit normal. The tangent angle of the urve is the angle  between the unit tangent and the positive x-axis. It is de ned as modulo 2. However, on e the tangent angle at a ertain point on the urve is spe i ed, a hoi e of ontinuous tangent angles along the

ow is determined uniquely. The urvature of with respe t to n, k , is de ned via the Frenet formulas, dt = kn; ds

dn = ds

k t;

(1.1)

where ds = j p jdp is the ar -length element. Expli itly we have k=

2 1 1 2

pp

pp p p j pj3 :

We shall study the ow  = F ( ; ; k)n; t

(p; t) 2 I  (0; T ); T > 0;

(1.2)

where F = F (x; y; ; q) is a given fun tion in R2  R  R , 2-periodi in . A ( lassi al) solution to (1.2) is a map from I  (0; T ) to R2 satisfying (i) it is ontinuously dierentiable in t and twi e

ontinuously dierentiable in p, (ii) for ea h t, p 7 ! (p; t) is a

urve, and (iii) satis es (1.2) where n and k are respe tively the unit normal and urvature of (
; t) with respe t to n. Given a urve

0 , we are mainly on erned with the following Cau hy problem:

To nd a solution of (1.2) whi h approa hes 0 as t # 0.

© 2001 by Chapman & Hall/CRC

1.1.

3

Short time existen e

For simpli ity, let's assume F is smooth in all its arguments. It is alled paraboli in a set E if F=q(x; y; ; q) is positive for all (x; y; ; q) in E and uniformly paraboli in E if there are two positive numbers  and  su h that 6

F q

6;

holds everywhere in E . Further, F is alled symmetri in E if F (x; y;  + ; q ) =

F (x; y; ; q ) ;

holds in E . When the set E is not mentioned, it is understood that F is paraboli , uniformly paraboli , or symmetri in its domain of de nition. Among the ows des ribed in the prefa e, the urve shortening ow is uniformly paraboli and symmetri . The generalized urve shortening ow is paraboli , symmetri but not uniformly paraboli in R3  R nf0g when  is not equal to 1. The anisotropi

urvature-eikonal ow is uniformly paraboli but not symmetri in general. This is lear from its physi al meaning: Reversing the orientation of the urve means inter hanging the phases. Although the initial phase boundaries are identi al and the dieren e in temperature is the same, the ows evolve in dierent ways. Re all that a reparametrization of a urve is another urve

(p) = ((p)) where  is a dieomorphism. The reparametrization is orientation preserving if  is positive and orientation reversing if  is negative. It is an important fa t that (1.2) is invariant under any orientation preserving parametrization. In fa t, an orientation preserving reparametrization (p; t) = ((p ); t) solves (1.2) if itself is a solution. Equation (1.2) is also invariant under any orientation reserving reparametrization when F is symmetri . In parti ular, for an embedded solution, that is, (
; t) is embedded for ea h t, the ow is independent of whi h parametrization is used in the beginning, and 0

0

0

0

© 2001 by Chapman & Hall/CRC

0

4

CH. 1.

Basi Results

so it only depends on the geometry of the initial urve. In this sense, (1.2) is geometri . However, this may no longer be valid for immersed

urves. In an example following the proof of Proposition 1.6, one will see an example whi h really depends on the parametrization of the initial urve. What is the type of (1.2)? At rst sight, the most natural way is to view it as a system of two equations for the two unknowns 1 and 2 . To examine its type, we need to linearize the system at a solution . Thus, let (") be a family of solutions of (1.2) satisfying

(0) = where is parameterized in ar -length. Then = d =d"(0) satis es  t

=

F  2 M 2 q p

+g ;

where

2 ( p2 )2 6 M =4

01 p2

p1 p2

( p1 )2

3 75

;

and g depends on , p but not on pp. It is lear that M is a non-negative matrix and has a unique null dire tion given by p , re e ting the invarian e of (1.2) under reparametrization. So even if F is paraboli , (1.2) is never paraboli when viewed as a system. To retrieve paraboli ity, we need to x a parametrization and express (1.2) as a single paraboli equation. There are several ways to a hieve this goal. The simplest one is to express the solution in lo al graphs. Suppose during some time interval ea h (
; t) is the graph of a fun tion u(x; t) de ned over some interval J . If ux is bounded, using the inverse fun tion theorem we an write (p; t) = (x; u(x; t))

© 2001 by Chapman & Hall/CRC

1.1.

5

Short time existen e

where x = x(p; t) and u is as regular as . We have  t

=

x u (1; ux ) + (0; ) : t t

p

Taking the inner produ t with n = ( ux ; 1)= 1 + u2x (here we assume the orientation of the urve is along the positive x-axis), we see that u satis es the equation u t

=

p1 + u

2 x

F (x; u; ; k ) ;

(1.3)

where now tan  = ux and k is given by k=

uxx

(1 + u2x )3=2

:

Clearly, when the gradient of u is bounded, (1.3) is a paraboli equation if and only if F is paraboli . Another useful way to obtain a single paraboli equation from (1.2) is to represent the ow as graphs over some xed urve. When

0 is smooth, we may take the xed urve to be 0 . But, it is always more onvenient to take it to be a smooth or analyti urve lose to

0 . For a xed losed, smooth urve near 0 , we an express the solution of (1.2) starting at 0 as

(p; t) =

(p) + d(p; t)N (p) ;

where N is the unit normal of . When t is small, d is small and dp is bounded. Let's write down the equation for d. To simplify the

omputation we shall assume is parametrized by the ar -length, i.e., j pj = 1. First, by (1.1),

t

=

p

j pj2 = (1

© 2001 by Chapman & Hall/CRC

+ dp N

j pj

d

T

d)2 + d2p ;

;

6

CH. 1.

Basi Results

where T and  are respe tively the unit tangent and urvature of . We also have k

=

(1 d)dpp + 2d2p + p ddp 22 d + 3 d2 : [(1 d)2 + d2p ℄3=2

(1.4)

It follows that d satis es dt

=

1 d F ( ; ; k ) : [(1 d)2 + d2p ℄1=2

(1.5)

When F is paraboli , (1.5) is paraboli as long as d < 1 and dp is bounded. Lo al solvability of the Cau hy problem for (1.5) an be readily dedu ed from standard paraboli theory. Before formulating an existen e result, we rst show the equivalen e between (1.2) and (1.5). In fa t, the following result holds. Proposition 1.1

Consider the ow  t

= F ( ; ; k)n + G( ; ; k)t ;

(1.6)

where F and G are smooth and 2-periodi in . Let be a solution of (1.6) in C (S 1  [0; T )). There exists ' : S 1  [0; T ) ! S 1 satisfying ' > 0 and '(p; 0) = p su h that (p; t) = ('(p; t); t) solves (1.2). 1

0

Proof:

0

With the notation as above, we have  0 (p; t) t

=

  ' ('; t) + ('; t) t p t

= F ( ;  ; k )n + G( ;  ; k )t + 0

© 2001 by Chapman & Hall/CRC

0

0

0

0

0

'

p : t

1.1.

7

Short time existen e

So solves (1.2) if 0

' = t

j p(')j G( ;  ; k): 0

(1.7)

0

With already known, (1.7) an be regarded as an ordinary dierential equation where p is a parameter. From the smooth dependen e on a parameter for solutions of ODE's, we dedu e the existen e of a solution ' satisfying '(p; 0) = p and ' > 0.  0

This proposition shows that the geometry of the ow (1.6) depends only on its normal velo ity F while the tangent velo ity merely alters the parametrization of the ow. Consider the Cau hy problem for (1.2) where F is smooth and paraboli , and 0 belongs to C 2; (S 1 ) for some  2 (0; 1). There exists a positive ! 6 1 su h that (1.2)
admits a solution in Ce2; S 1  [0; !) satisfying (
; 0) = 0 . Moreover, the followings hold: Proposition 1.2 (lo al existen e)

(i) is smooth (and analyti if F is analyti ) in S 1  (0; !), (ii) if ! is nite, then k(
; t), the urvature of (
; t), be omes unbounded as t " !, and (iii) if 0 depends smoothly (or analyti ally and F is analyti ) on a parameter, so does .

From now on, we all the solution de ned in (0; !) the maximal solution of (1.2) starting at 0 . Applying Fa t 3 in the next se tion to (1.5) and then using Proposition 1.1, we know (1.2) has a unique solution in C S 1 
[0; T ℄ , T > 0, (
; 0) = 0 , where T depends on the C 2;-norm of

0 . Moreover, (i) and (iii) hold. Also, the solution an be extended as long as the C 2; -norm of is under ontrol for some 2 (0; 1). Proof:

1

© 2001 by Chapman & Hall/CRC

8

CH. 1.

Basi Results

We shall show that a uniform urvature bound yields a bound on some C 2; -norm for . In fa t, we an always use a rigid motion to bring a point (p0 ; t0 ) to the origin so that n(p0 ; t0 ) = (0; 1). The urvature bound ensures that there exist Æ > 0 and a re tangle R = ( a; a)  ( b; b) su h that, for all t 2 (0; !), jt t0 j 6 Æ, T

(
; t) R is the graph of a fun tion u(x; t) whose rst and se ond derivatives in x are bounded in ( a; a). By dierentiating (1.3), we see that the fun tion w = ux satis es a uniformly paraboli equation of the form wt = a(x; t)wx


x + b(x; t)wx + (x; t)w + f (x; t)

;

where the oeÆ ients and f are uniformly bounded. By Theorem 11.1 in Chapter 3 of Ladyzhenskaja-Solonnikov-Ural eva [86℄, we on lude that there exists some 2 (0; 1) su h that kukCe2; is uniformly bounded in any ompa t subset of R  (t0 ; t0 + Æ). By paraboli regularity theory, all higher derivatives of u are bounded in any ompa t subset of R  (t0 ; t0 + Æ) as well. If the urvature of (
; t) is bounded near !, we may solve (1.2) using (
; t0 ) where t0 is lose to ! as initial urve to obtain a solution whi h extends beyond !. This is impossible. Hen e, the urvature must be ome unbounded as t " !.



The above proof has established the following estimate: Let be a solution of (1.2) in S 1  [0; T ). Then, for any Æ > 0 and k > 1, there exists a onstant C depending on the urvaRemark 1.3

ture su h that

max

on

n  i+j i j

: i + 2j 6 k

s t S 1  [0; T ), where s = s(t)

o

6C ;

the ar -length parameter of

(
; t).

To see this, let's rst observe that it follows from the above proof that k=s are bounded by k in [0; T ). Next, by dierentiating

© 2001 by Chapman & Hall/CRC

1.1.

9

Short time existen e





the relations 2 = 1 and 1 2 2 1 = k repeatedly we see that  =s an be estimated by k and its derivatives up to (n 1)-th order. Finally, we an estimate the derivatives of in t by dierentiating (1.2). s

n

s

ss

ss

s

n

Proposition 1.4 (uniqueness) Let 1 and 2 be two solutions of

 [0; T )) where F is paraboli . If 1 (
; 0) = 2(
; 0) in some parametrization, then 1 (
; t) = 2 (
; t) for all t.

e 0;1 (S 1 (1.2) in C

Represent 1 and 2 as graphs over a smooth urve lose to

1 (
; 0) and then use Fa t 6 to dedu e that they are identi al for all t in [0; T ). Note that representing a losed urve as a graph over some smooth urve is possible provided the urve is Lips hitz ontinuous.

Proof:



Now we formulate two useful properties of the ow (1.2). They are immediate onsequen es of the strong maximum prin iple. Proposition 1.5 (preserving embeddedness) where

F

Consider (1.2)

is paraboli and symmetri . Then, any solution

(
; t)

in

 [0; T )) is embedded if 0 is embedded. Proof: Sin e (
; t) tends to 0 in C 0 1 -norm as t # 0, (
; t) is embedded for all small t. Suppose that (
; t) has a self-interse tion at some time. We an nd p; q, p = 6 q, and t1 > 0 su h that (p; t1 ) =

(q; t1 ), but (
; t) is embedded for all t less than t1 . Without loss Ce 0;1 (S 1

;

of generality we may take (p; t1 ) = (0; 0) and n(p; t1 ) = (0; 1). In a T suÆ iently small re tangle R = ( a; a)  ( b; b), R (
; t1 ) is the union of two graphs u ; i = 1; 2; over ( a; a) satisfying u1 (0; t1 ) = u2 (0; t1 ), u1 (0; t1 ) = u2 (0; t1 ) and u2 > u1 at t = t1 . Moreover, u2 > u1 for all t < t1 and lose to t1 . By the embeddedness of

(
; t); t < t1 , (x; u2 (x; t)), say, is transversed along the negative xaxis and (x; u1 (x; t)) along the positive x-axis. However, sin e F is i

x

© 2001 by Chapman & Hall/CRC

x

10

CH. 1.

Basi Results

symmetri , reversing the dire tion of (x; u2 (x; t)) does not hange the equation it satis es. So both u1 and u2 solve the same equation, and we may use the strong maximum prin iple to on lude that u2 = u1 for all t < t1 , and the ontradi tion holds.  Proposition 1.6 (strong separation prin iple) Consider (1.2) where

F

is paraboli and symmetri .

tions of (1.2) in

C ([0; T ); C 0;1 (S 1 ))

Let

and let

1

and

D1 (t)

2

be two solu-

be a omponent of

n 1 (
; t) whose boundary D1 (t) hanges ontinuously in t. Suppose that 2 (
; 0) is ontained in D1 (0) and tou hes D1 (0) only at regular points of 1 (
; 0) or is disjoint from D1 (0). Then, 2 (
; t) is R

2

ontained in

D1 (t)

for all

t > 0.

A point on a urve is alled a regular point if there is a disk D T

ontaining this point su h that D (
) is a one-dimensional manifold. Parametrize both 1 (
; 0) and 2 (
; 0) by ar -length. We shall T prove the proposition when 2 (
; 0) D1 (0) is non-empty. Let X T be a point on 2 (
; 0) D1 (0). By our hypotheses, we an nd a maximal interval [ a; a℄ on whi h 2 (
; 0) and 1 (
; 0) oin ide. Now represent 2 (
; 0) and 1 (
; 0) as graphs over a smooth urve de ned in J = [ a Æ; a + Æ℄ for some small Æ. We may assume that d2 (
; 0) > d1 (
; 0) on J and d2 ((a + Æ ); t) > d2 ((a + Æ ); t) for all small t. By Fa t 6, d2 > d1 for t > 0. In other words, 2 is separated from 1 instantly. A similar argument shows that they annot tou h afterward.  Proof:

At this point, we present an example showing that the ow (1.2) in general, depends on the parametrization of the initial urve. Let C be the ir le fx2 + y 2 = 4g and C the ir le fx2 + (y + 1)2 = 1g. We may modify both ir les near A = (0; 2) so that the resulting 0

© 2001 by Chapman & Hall/CRC

1.1.

11

Short time existen e

urves have smooth onta t at A. Let B = (0; 2) and B = (0; 0). Consider two urves: 1 is ABAB AB A and 2 is ABABAB A, both parametrized by ar -length and transversing in ounter lo kwise dire tion. Mark a point P and a point Q on C whi h lie on the left and the right of A, respe tively. Consider the ar P ABAQ on 2 . We an nd a orresponding ar P ABAQ on 1 with the same length. When P and Q are lose to A, we an represent the se ond ar as a graph over the rst ar . So d1 (
; 0) > d2 (
; 0)  0 and d1 (
; t) > d2 (
; t) for small t at the endpoints. By Fa t 6, the ar s will be separated instantly. It shows that, although the geometry of

1 and 2 are the same, their ows are dierent. Now we give some onsequen es of the Sturm os illation theorem (see Fa t 7 in the next se tion). 0

0

0

0

0

0

Let 1 and 2 be two solutions of (1.2) in C (S 1  (0; T )) where F is paraboli and symmetri . Denote the number of interse tion points of 1 (
; t) and 2 (
; t) by Z (t). Then Z (t) is nite for all t in (0; T ), and it drops exa tly at those instants t when 1 (
; t) and 2 (
; t) tou h tangentially at some point. Moreover, all these instants form a dis rete subset of (0; T ). 1

Proposition 1.7

Proof:

Use the lo al form (1.3) and Fa t 7 in the next se tion.



Let be a solution of (1.2) in C (S 1  (0; T )) where F is paraboli . Denote the number of nodes of (
; t) by N (t). Then N (t) is nite for all t in (0; T ) and drops exa tly at those instants when F ( (
; t); (
; t); k(
; t)) has a double zero. The instants at whi h this happens form a dis rete set.

Proposition 1.8

1

A point (p) on a urve is a node of F if F ( (p); (p); k(p)) = 0. In the urve shortening ow, a node is simply an in e tion point of the urve.

© 2001 by Chapman & Hall/CRC

12

CH. 1.

Basi Results

For ea h xed t , we may represent (
; t) as graphs over

(
; t ) for all t lose to t . From (1.4), we know that a node of (
; t) is a zero of dt (
; t). By dierentiating (1.4), dt is seen to satisfy a linear paraboli equation, and so the proposition follows from Fa t 7. 

Proof:

We end this se tion by establishing a more general existen e result for immersed urves. Consider the Cau hy problem for (1.2) where F is uniformly paraboli , symmetri , and satis es (i) for any ompa t K in R2  S 1  R, there exists a onstant C su h that Proposition 1.9

















F F F F 2 + + + jq j 6 C (1 + q ) ; x y  q

(1.8)

for all (x; y; ; q) in K  R, and (ii)

F ( ; ; 0) only depends on  :

(1.9)

Then, for any losed C 0;1- urve 0 , there exists T > 0 su h that the Cau hy problem for (1.2) has a unique solution in Ce0;1 (S 1  [0; T )) \ C 1(S 1  (0; T )).

Sin e 0 is a C 0;1- urve, we an nd open intervals J ;  = 1;


; N , in dierent oordinates and C 0;1-fun tions u0 de ned over J su h that 0 an be des ribed as the union of the graphs f(x; u0 (x)) : x 2 J g,  = 1;


; N . Let f 0j g be a smooth approximation to 0  where 0j is the graph of some smooth u;j 0 in J onverging to u0 in C 0;1 -norm. By Proposition 1.2, for ea h j there is a maximal solution

j taking 0j as its initial urve in [0; !j ); !j > 0. We rst show that j f! g has a uniform positive lower bound. Let C0 be any ir le of radius Æ entered at a point lying on the 3Æ-neighborhood of 0 and let C (t) be the ow (1.2) starting at Proof:

© 2001 by Chapman & Hall/CRC

1.1.

13

Short time existen e

C0 . There is a uniform, positive t1 su h that all C (t) exist and stay outside the Æ-neighborhood of 0 for all t in [0; t1 ℄. By the strong separation prin iple, we know that, for suÆ iently large j , j (
; t) is

on ned to the Æ-neighborhood of 0 for t < minf!j ; t1 g. On the other hand, for ea h  and x 2 J , onsider the verti al line segment `x whi h passes the point (x; 0) and whose endpoints lie on the boundary of the 4Æ-neighborhood of 0 . Sin e F ( ; ; 0) only depends on , the ow (1.2) starting at `x , `x (t), is again a verti al

line segment, and it moves horizontally at onstant speed. For ea h , x a subinterval J0 , J0 J , so that the union of the graphs (x; u;j (x; t)) over J0 still overs to j ( ; t). There exists t2 > 0 su h that the line segments `x (t); x J , over J0 for all t < t2 . Besides, the endpoints of `x (t) lie outside the 2Æ-neighborhood of

0 . Now, as ea h `x interse ts the graph of u;j 0 transversally at exa tly one point, by the Sturm os illation theorem (Fa t 7 in the next se tion) the graph of u;j ( ; t) annot be ome verti al in J0 [0; t3 ℄, t3 = min !j ; t1 ; t2 . This means that the gradient of u;j is bounded. Observe that u u;j satis es (1.3), whi h is now written as 




2






f

g



ut =

p1 + u

2

x

(x; u; ux ; uxx ) :

So w u;j =x satis es 

p

wt = ( 1 + w2 (x; u(x; t); w; wx ))x :

By uniform paraboli ity and the stru tural ondition (1.7), it follows from Theorem 3.1 in Chapter 5 of [86℄ that wx is uniformly bounded on every ompa t subset of J0 (0; t3 ℄. Thus, the urvature of ea h

j is bounded as long as t 6 t = min t1 ; t2 . By Proposition 1.2,

j exist in [0; t ℄ for all large j . Next, we derive a uniform gradient estimate for u;j by the following tri k. Instead of using `x , we tilt it a little bit to the right 

f

© 2001 by Chapman & Hall/CRC

g

14

CH. 1.

`0x and `x . Ea h `0x (resp. `x )  00 > =2) with the positive x-axis. 00

and to the left to obtain line segments makes an angle

 0 < =2

When

are lose to

0

 00

and

(resp.

Basi Results

=2, `0x

and

00

`x

00

are still transversal to the

;j graph of u0 , and their endpoints still lie outside the Æ -neighborhood

of

0 .

By the Sturm os illation theorem again,

juxj 6 maxftan 0; j tan 00 jg 0  [0; t ℄. As before, it implies a uniform grafor all u = u;j over J 0  (0; t ℄ and, by paraboli dient bound on every ompa t subset of J regularity, all higher order bounds follow. By passing to a onvergent subsequen e, we obtain a solution of (1.2) whi h is a lo al Lips hitz graph of

2 J0 ; t 6 t with uniform C 0;1-norm.

u (x; t), x

tion 1.10 below ensures that this solution takes

e0;1 (S 1 and it belongs to C Proposition 1.10

Let

 [0; t ℄).

0

Proposi-

as its initial urve



be a solution of (1.2) in [0; T ) where

uniformly paraboli and symmetri . Suppose that

F

f (
; t) : t 2 [0; T )g

is ontained in some bounded set. There exist positive onstants and

C

depending on

F

C1

satisfying 0

6t

(1.10)

t0 < minf; T g.

By uniform paraboli ity, there exist positive onstants

Proof:

and

t



and the bounded set su h that

(
; t)  NC pt t0 ( (
; t0 )) for all t0 and

is

k0

su h that

fF (x; y; ; q) : (x; y) 2 D and  2 Rg 6 C1 q

sup

> k0 , where D is a bounded set ontaining the image of

(
; t), for all t 2 [0; T ). Let C0 be any ir le of radius Æ , Æ 6 1=k0 ,

entered at the boundary of the 2Æ -neigborhood of (
; t); t0 2 [0; T ), for all

q

and denote the solution of

C t

=

© 2001 by Chapman & Hall/CRC

C1 k n;

C (
; t0 ) = C0 ;

1.2.

15

Fa ts from the paraboli theory

by C (t). This solution is a family of shrinking ir les whose radius at p t is given by Æ2 2C1 t. Hen e, it exists for t in [t0 ; t0 + Æ2 =(2C1 )). Noti ing that the urvature of C (t) is in reasing and so (1.10) ontinues to hold, we infer from Proposition 1.6 that (
; t) and C (t) are disjoint. Sin e the enter of C (t) ould be any point on the boundary of N2Æ ( (
; t0 )), (
; t) stays inside N2Æ ( (
; t0 )) as long as 0 6 t t0 < Æ2 =(2C1 ). In parti ular, taking t t0 = Æ2 =(4C1 ) yields  the desired result where C = (ÆC1 )1=2 and  = (4C1 k02 ) 1 . 1.2

Fa ts from the paraboli theory

Let E be a region in R2 . We onsider the equation ut = (x; t; u; ux ; uxx ) ; (x; t) 2 E ;

(1.11)

where (x; t; z; p; q) is smooth in E  R3 . This equation is paraboli if  =q is positive in E  R3 , and uniformly paraboli if  6  =q 6  holds for some positive onstants  and . A fun tion de ned in E is alled a ( lassi al) solution of (1.11) if it is ontinuously dierentiable in t, twi e ontinuously dierentiable in x and satis es (1.11). From now on, a solution always means a lassi al solution. The paraboli Holder spa e is de ned as follows. Let Q = I  (0; T ) be a ylinder. For a fun tion u de ned in Q, we introdu e the following semi-norms and norms: for ea h integer k > 0 and  2 (0; 1℄, set  

u  = sup

n

kukCek; Q = (

)

© 2001 by Chapman & Hall/CRC

ju(x; t) u(y; s)j : (x; t); (y; s) 2 Qo ; (jx yj + jt sj)= 2

2

X h  i+j u i  i+j u

+

i tj C (Q) i tj  : x x i+2j 6k i+2j =k X

16

CH. 1.

Basi Results

The paraboli Holder spa e Cek;(Q) is the ompletion of C (Q) under k
kCek; (Q) . We rst state results for the linear equation 1

ut = a(x; t)uxx + b(x; t)ux + (x; t)u f (x; t) :

(1.12)

a priori estimates) Consider (1.12) in S  (0; T ) where it is uniformly paraboli and the oeÆ ients and f are 2-periodi and in Cek;(Q) for some (k; ). Then, for any solution u in Cek ;(Q), there exists a onstant C whi h depends on ; ; k;  and the Cek;norms of the oeÆ ients su h that 1

Fa t 1 (

+2



u ek+2; C

(





6 C f Cek; Q + u(
; 0) C k+2; S1 :

Q)

(

)

(

)

Moreover, for any sub- ylinder Q = S  (t ; T ), t > 0 there exists a onstant C whi h depends on t ; ; ; k;  and the Cek;norms of the oeÆ ients su h that 1

0

0



u ek+2; C

(

Q) 0

0

0

0





6 C f Cek; Q + u(
; 0) C S1 : 0

(

)

(

)

(1.13)

Consider (1.12) in S  (0; T ) where it is uniformly paraboli , and the oeÆ ients and f are 2-periodi and in Cek;(Q) for some (k; ). Then, for any u in C k ;(S ), there exists a unique solution u in Cek ; (Q) satisfying u(
; 0) = u . 1

Fa t 2 (existen e and uniqueness)

0

+2

1

+2

0

Fa t 2 may be proved in the following way. First, solve the Cau hy problem in the smooth ategory by means of separation of variables. Then use an approximation argument oupling with the global a priori estimate in Fa t 1 to get the general result. Now we onsider the nonlinear equation (1.11).

© 2001 by Chapman & Hall/CRC

1.2.

17

Fa ts from the paraboli theory

S 1  (0; T ) where  is smooth and paraboli . For any u0 in C k+2; (S 1 ) for some (k; ), there exists a positive t0 6 T su h that (1.11) has a solution u in Cek+2;(Q) satisfying u(
; 0) = u0 . Moreover, if u0 depends smoothly Fa t 3 (lo al solvability) Consider (1.11) in

on a parameter (resp. analyti ally on a parameter and lyti ), then

u also depends smoothly (resp.



is ana-

analyti ally) on the same

parameter.

Fa t 3 is a onsequen e of the inverse fun tion theorem. In fa t, by repla ing u by u u0 , we may assume the initial value is identi ally zero. Let Q(t) = S 1  (0; t), t 6 t0 where t0 is to be hosen later,  and de ne a map F from X = u 2 Cek+2;(Q(t)) : u(
; 0) = 0 to Cek;(Q(t)) by F (u) = ut (x; t; u; ux ; uxx) : The Fre het derivative of F at the spe i ed fun tion u0 = (x; t; 0; 0; 0)t is given by DF (u0 )v = vt

 

q

vxx +

   vx + v ; p z

where the oeÆ ients are evaluated at u0 . Sin e  is paraboli , it follows from Fa t 2 that DF (u0 ) is invertible. By the inverse fun tion theorem there exist t0 ; ; Æ su h that, for any f satisfying kf F (u0 )kCek;(Q(t)) < Æ, there exists a unique u satisfying ku u0 kCek+2; (Q(t)) < Æ su h that F (u) = f for all t 6 t0 . As F (u0 ) tends to zero as t # 0, there is some t1 > 0 su h that kF (u0 )kCek; (Q(t)) < Æ. In other words, F (u) = 0 is solvable in Cek+2;(S 1  [0; t1 ℄). Now, the smoothnanalyti dependen e on a parameter is a onsequen e of the impli it fun tion theorem. For an appropriate version see, e.g., Zeidler [112℄.

© 2001 by Chapman & Hall/CRC

18

CH. 1.

Basi Results

Let u 2 C ;(Q) be a solution of (1.11) where  is smooth and paraboli . Then u belongs to C 1(S  (0; T )). Moreover, if  is further analyti , then u is analyti in S  (0; T ). n

e2

Fa t 4 (instant smoothness analyti ity) 1

1

This fa t an be dedu ed from Fa t 3. For any solution u, we set ua;b = u(x + at; bt) where a is near 0 and b near 1. Then ua;b solves (1.11) for  = a;b  aux + b(x + at; bt; u; ux ; uxx ), whi h depends smoothlynanalyti ally on (a; b). A

ording to Fa t 3,  j +k ua;b aj bk (a;b)=(0;1)

j +k

 u = tj +k x j tk (x;t) :

Hen e, u is smoothnanalyti for t > 0. Next we onsider the strong maximum prin iple. Re all that a ( lassi al) subsolution (resp. supersolution) of (1.11) or (1.12) is a fun tion whi h satis es (1.11) or (1.12) with \=" repla ed by \6" (resp. \>"). For any (x0 ; t0 ) in E , we de ne E (x0 ; t0 ) to be the set

onsisting of all points in E whi h lie on or below the horizontal line segment ` passing through (x0 ; t0 ) in E or an be onne ted to ` by a verti al line segment ontained inside E .

Consider (1.12) where it is uniformly paraboli , the oeÆ ients are bounded, 6 0 and f = 0. Suppose u is a subsolution of (1.12) whi h attains a non-negative maximum M at (x ; t ) in E . Then u = M in E (x ; t ). Fa t 5 (strong maximum prin iple)

0

0

0

0

Let u and v be, respe tively, a subsolution and a supersolution of (1.11) whi h is paraboli , in C (Q). Suppose that u 6 v on the paraboli boundary pQ  I  f0g I  [0; T ). Then, either u  v or u < v in Q n p Q.

Fa t 6 (strong omparison prin iple)

S

Let w = et (u v). Then, for suÆ iently large , w satis es a

© 2001 by Chapman & Hall/CRC

1.3.

19

The evolution of geometri quantities

uniformly paraboli , linear equation of the form (1.12) where 6 0. So Fa t 6 follows from Fa t 5. In Chapter 8 we need a weak version of Fa t 6. Let's onsider (1.12) where a is non-negative, the oeÆ ients are bounded and f 0. The weak maximum prin iple states that a subsolution whi h is non-positive on the paraboli boundary of E is non-positive in E . Using this prin iple, we immediately dedu e the weak omparison prin iple: Let u and v be a subsolution and a supersolution of (1.11) in C (Q), respe tively. Suppose that  =q > 0 and u 6 v on P Q. Then u 6 v in Q. Finally, we need the following: 

Fa t 7 (\Sturm os illation theorem") Consider (1.12) in Q where it is uniformly paraboli , a; ax ; axx ; at ; b; bx ; bt and are bounded measurable, and f = 0. Suppose u is a solution whi h never vanishes on I  [0; T ℄. Then Z (t), the number of zeroes of u(
; t), is nite and non-in reasing for all t 2 (0; T ) and it drops exa tly at multiple zeroes. The set ft 2 (0; T ) : u(
; t) has a multiple zero g is dis rete.

1.3

The evolution of geometri quantities

In this se tion, we derive the evolution equations satis ed by the basi geometri quantities for the solution of (1.6). We rst ompute the evolution of the ar -length element s = p . By (1.1), we have j

s2 t

© 2001 by Chapman & Hall/CRC

 

= 2

  ; p t p

   = 2 p ; p

 t

 

j

20

CH. 1.

=

Basi Results

2F s2 k + 2s2 Gs:

Hen e s = ( F k + Gs)s : t

(1.14)

Next, the unit tangent t satis es t = t

1 s 

s2

t p

+

1 

s p

(F n + Gt )

= (Fs + Gk)n: Now, n = t

 n   n  ;n n + ;t t t t (Fs + Gk)t:

=

By dierentiating both sides of t = ( os ; sin ) we have  = Fs + Gk: t

(1.15)

By the Frenet formulas, we know that k = =s. Therefore, k t

 = t

=

1

 s p



1 s 

s2

= (F k

t p

+

1 

 

 s p t

Gs )k + (Fs + Gk )s :

(1.16)

In parti ular, k = kss + k3 t

© 2001 by Chapman & Hall/CRC

(1.17)

1.3.

21

The evolution of geometri quantities

holds for the CSF. The length of (
; t); L(t), satis es dL = dt

= =

Z

s dp t

I

Z Z

(F k

Gs )ds

(1.18)

F kds:

So, the urve shortening ow is the negative L2 -gradient ow for the length. For the generalized urve shortening ow for losed urves, we have Z dL jkj1+ ds: = dt

So, the length of the urve is stri tly de reasing along the ow. In fa t, by Z jkjds > 2

and the Holder inequality, we have dL dt

6

(2)+1 L



;

and L1+ (t0 ) 6 L1+ (0)

(2)1+ (1 + )t :

It implies an upper bound on the life span whi h only depends on the length of the initial urve.

© 2001 by Chapman & Hall/CRC

22

CH. 1.

Basi Results

For an embedded losed solution we an use the formula for the en losed area A=

1 2

Z

< ; n > ds

to ompute 1 2

dA = dt

1 + 2

Z

1 F ds + 2

Z

Fk

Z




Fs + Gk < ; t > ds




Gs < ; n > ds ;

whi h, after integration by parts, gives dA = dt

Z

F ds :

(1.19)

For the urve shortening ow, we have the following remarkable property: A(t) = A(0)

2t :

(1.20)

It says that, no matter what the initial urve is, the life span of the

ow is equal to A(0)=2, whi h only depends on the initial area. (In the next hapter, we shall show that the ow exists until A(t) = 0.)

Notes Lo al existen e. It was Gage and Hamilton [58℄ who rst pointed out the weak paraboli ity of the urve shortening ow when it is viewed as a system for ( 1 ; 2 ). Instead of redu ing it to a single paraboli equation, they employed a version of the Nash-Moser impli it fun tion theorem to show lo al existen e. However, in subsequent work, lo al existen e was obtained via redu tion to a single

© 2001 by Chapman & Hall/CRC

1.3.

23

The evolution of geometri quantities

equation in various way. In [13℄ and [14℄, Angenent studied (1.2) on a surfa e. Very general results on lo al existen e and the existen e of limit urves are obtained. In parti ular, Propositions 1.2{1.10 are essentially ontained in [14℄. Some of his main results on lo al existen e may be des ribed as follows. Let M be a two-dimensional Riemannian manifold and let (
; t) be a family of urves, from S 1 to M . At ea h point of the urve, we an de ompose the velo ity ve tor t into its normal and tangent

omponents with respe t to a frame (t; n). Consider the ow V

= F (t; k) ;

(1.21)

where V is the normal velo ity of (
; t). The fun tion F is de ned in S 1 (M )  R where S 1 (M ) is the unit tangent bundle of M and t; k are, respe tively, the tangent and urvature of (
; t). For simpli ity, F is assumed to be smooth in S 1 (M )  R, and it further satis es some of the following assumptions: for some positive ; ;  and  , (i)  6 F=k 6 , F (t; 0)

6 ,
(iii) rhF + krv V 6  1 + jkj2 , (ii)

(iv) F ( t;

k) =

and

F (t; k ),

in S 1 (M )  R. Here rh and rv are, respe tively, the horizontal and verti al omponents of the gradient of F , rF , in t. Now we an state: Theorem A Under (i){(iii), the Cau hy problem for (1.21) has a maximal solution in (0; ! ); ! > 0, for any lo ally Lips hitz 0 . Theorem B

Under (i){(vi), the Cau hy problem for (1.21) has a

maximal solution in

0 .

© 2001 by Chapman & Hall/CRC

(0; !); ! > 0, for any C 1 -lo ally graph-like urve

24

CH. 1.

Basi Results

See [14℄ for the de nition of a C 1 -lo ally graph-like urve. We point out that any urve whi h is lo ally the graph of a ontinuous fun tion is C 1 -lo ally graph-like. In parti ular, it implies that Proposition 1.9 holds without (1.8).

Paraboli theory. We refer to the books [86℄, Protter-Weinberger [96℄, and Lieberman [88℄ for detailed information on the basi fa ts of paraboli theory, ex ept for Fa t 7. The dedu tion of Fa t 4 from Fa t 3 is taken from [13℄ where it is attributed to DaPrato and Grisvard. Fa t 7, the most general form of the Sturm os illation theorem for paraboli equations, is taken from Angenent [12℄. Results of this type were established by many authors sin e Sturm in 1836.

Geometri ows. The equivalen e between (1.2) and (1.6) was formulated in Epstein-Gage [48℄. Here our proof follows Chou [29℄. Usually, people hoose G to vanish identi ally. Another useful hoi e is to make the parametrization have onstant speed at ea h t. We shall illustrate this point below. Apparently, Proposition 1.1 holds when F and G depend also on the derivatives of k with respe t to the ar -length. Equation (1.2) in this form may be alled a geometri evolution equation. Quite a number of geometri evolution equations have been studied in re ent years. Usually, they are losely related to geometri fun tionals, su h as the area, the length, or some urvature integrals. For instan e, the generalized urve shortening ow de reases both the area and the length of an embedded, losed urve. In some physi al situations, one would like to have a ow whi h de reases the length but preserves the area (the total mass). The simplest hoi e of paraboli ows of

© 2001 by Chapman & Hall/CRC

1.3.

25

The evolution of geometri quantities

this kind is the ow driven by surfa e diusion  = t

kss n :

The reader is referred to Cahn-Taylor [24℄, Cahn-Elliott-Novi k-Cohen [23℄ and Giga-Ito [64℄ for physi al ba kground and analysis of this

ow. Flows de reasing the elasti energy Z

k 2 ds

are also studied by some authors. By taking variations in dierent fun tion spa es, one arrives at three dierent urvature ows assoi ated to this energy, see Langer-Singer [87℄, Wen [110℄, and [111℄. When one does not insist on paraboli equations and looks for

ows whi h preserve length and area, one may onsider F depending on the derivatives of the urvature in odd orders. As the simplest example, let's take F = ks. Clearly, the resulting ow preserves length and area. Let (
; t) be a solution of this ow. We
reparametrize it by setting (
; t) = p(s; t); t where s = s(p; t) is the ar -length element of (
; t). Then (
; t) satis es 0

0

 0 = t

ks n + Gt :

G an be determined by requiring s=t = 0. On one hand, we have 0

ts =

kss + Gk




n+

ks k + Gs




t

by (1.1). On the other hand, we have 0

st =

kss + Gk




n;

by (1.15). Sin e s=t = 0, we have ts = st . It implies that G = k 2 =2, and so the ow is given by 1 2

t = ks n k t: (1.22) 2 0

0

© 2001 by Chapman & Hall/CRC

0

26

CH. 1.

Basi Results

Now, by (1.16), the urvature k(s; t) satis es the modi ed K dV equation, kt

+ ksss +

3 2 k k = 0 2 s

:

(1.23)

This equation is one among many integrable equations studied extensively in the past several de ades. Sin e a urve is determined by its urvature up to a rigid motion, there is a formal equivalen e between (1.22) and (1.23). It turns out that many integrable equations are asso iated to the motion of urves in the plane or the spa e in this way. For further dis ussion, see Goldstein-Petri h [69℄ and Nakayama-Segur-Wadati [91℄.

© 2001 by Chapman & Hall/CRC

Chapter 2

Invariant Solutions for the Curve Shortening Flow We dis uss invariant solutions { travelling waves, spirals and selfsimilar solutions { for the generalized urve shortening ow (GCSF)   = k t

1

kn ;  > 0 :

(2.1)

They are not merely examples. In fa t, they an be used as omparison fun tions to yield a priori estimates. Even more signi ant is their role in the lassi ation of the singularities of (2.1). For the

urve shortening ow, the travelling waves, whi h are alled grim reapers, des ribe the asymptoti shape of type II singularity and the

ontra ting self-similar solutions (they are ir les when embedded)

hara terize type I singularities. We shall dis uss this in Chapter 5. Travelling waves, spirals, and expanding self-similar solutions appear to be very stable. They an be used to des ribe the long time behaviour of omplete, unbounded solutions of (2.1). 2.1

Travelling waves

A travelling wave is a solution of (2.1) whi h assumes the form v (x; t) = v (x) + t when it is expressed as a graph lo ally over some 27 © 2001 by Chapman & Hall/CRC

28

CH. 2.

Invariant Solutions for the Curve Shortening Flow

x-axis. It satis es the equation vxx = 1 + vx2


3

2

1

;

(2.2)

after a s aling in (x; t) to take = 1. It is easy to see that any solution of (2.2) is even and onvex. If we rotate the axes by 90Æ , the graph (x; v(x)) onsists of two bran hes of the form x (v t) and thus justi es the name of a travelling wave. Depending on the value of , these solutions an be put into two

lasses. First, for 0 <  6 1=2, v is entire over R and satis es x

and

 1 2  1 v(x) 1 = !1 x 11 2 

lim

 

2

  ; 1 


1=2, there exists a nite x depending on  su h that dv=dx blows up at x. Moreover,

lim

v(x) = 1 ( = 1) ; log(x x)

lim x"x

v(x) x"x (x x) 11

lim



2

=

 2



1  11

 

2

   1 1, travelling waves are not omplete as urves. Travelling waves with speed are given by the graphs of 1 v( x). Intuitively speaking, the speed in reases as the width of the wave narrows upward.

© 2001 by Chapman & Hall/CRC

2.2.

29

Spirals

2.2

Spirals

Spirals are travelling waves in the polar angle . To des ribe its equa
tion, we express the solution urve as r os((r) + t , r sin (r) +


t) , where the distan e to the origin r is the parameter of the urve. The resulting solution is a urve rotating around the origin with speed j j. For simpli ity, in the following dis ussion we always take

to be positive so that it rotates in ounter lo kwise dire tion. By a dire t omputation, the urvature of this urve is given by 2 (r) + r (r) + r2  3 (r) k (r ) = ; (1 + r2  2 )3=2 0

00

0

0

and the normal velo ity is given by

r  :
n = t (1 + r2  2 )3=2 0

Letting (r) = '(y), r = y2 , a solution of the GCSF is a spiral if and only if ' satis es

 d' = ; dy 2y

(2.3)

h 1 1 d = (1 + 2 ) y 2 dy 2

1 2

1 + 2


1

i : y

1 2

2

(2.4)

(2.3) and (2.4) an be ombined to yield tan

1




1 (y) 0 = y 2

2

1 2

1 + 2


1

2

1 2

'0 (y) ;

that is,

'(y) = tan

1

1 (y) + 2

Z y

1

z 2

1 2

1 + (z )2


1

2

1 2

In parti ular, when  = 1,

'(y) = y tan 1 (y) + onstant : 2

© 2001 by Chapman & Hall/CRC

dz :

(2.5)

30

CH. 2.

Invariant Solutions for the Curve Shortening Flow

We analyze (2.4) as follows. Setting its right hand side to zero, we have 1 1 

y 2 + 2 = 1 1 : (1 + 2 ) 2 2 This equation de nes a urve = (y; (y)) in (0; 1)  R, where  is stri tly in reasing and satis es (y) lim 1 + 1 = 1 ; (2.6) y#0 y 2 2 and y

(y) lim !1  y 2 + 21 = 1 :

(2.7)

The urve divides (0; 1)  R into an upper and a lower region. Any integral urve of (2.4) starting from the upper region stri tly de reases as y in reases until it hits ; then it be omes in reasing and tends to  asymptoti ally, never rossing  again. On the other hand, any integral urve starting from (0; 1)  ( 1; 0) is stri tly in reasing and tends to  from below as y ! 1. Denote by + the union of all integral urves starting in the upper region, and by

the union of all integral urve starting in (0; 1)  ( 1; 0). Both regions are open and + lies above . Set   (y) = inf  : (y; ) 2 + ; and 

 (y) = sup  : (y; ) 2



:

By ontinuity, both  and  are solutions of (2.4) satisfying  (0) =  (0) = 0. Using (2.4), it is not hard to see that (y) =


 12 + 21 y 1 + o(1) at y = 0 ; 2 + 1

(2.8)

for any solution  starting at the origin. It follows from (2.8) that the solution satis es (0) = 0 is = unique. Hen e,  and  are

© 2001 by Chapman & Hall/CRC

2.2.

31

Spirals

identi al. From now on we denote it by 0 . We all a (maximal) solution + (resp.  ) from + (resp. ) a type{I (resp. type{II) solution. It is not hard to see that, for any + or  , there exists Æ > 0 su h that lim  + (y ) = 1 y #Æ lim  (y ) = 1 : y #Æ

(2.9)

Moreover, for ea h Æ > 0 there exist a unique type{I and a unique type{II solution whi h blow up at Æ. The graphs of all type{I, type{II solutions, and 0 form a foliation of (0; 1)  R. The solution urve  = '(y) + t depends on two parameters be ause ' satis es a se ond order ODE. One of the parameters is the hoi e of  and the other is the integration onstant arising from integrating (2.3). The latter a

ounts for a rotation of the solution

urve in the (x; u)-plane and is not essential. We shall always assume that the solution urve starts at the positive x-axis. The solution

urve rotates in unit speed with its shape un hanged. It suÆ es to look at the snapshot at t = 0, i.e., '(y). First of all, let ' be a solution of (2.3) where  2 (Æ; 1), Æ > 0. Its urvature is given by 1

r  : k (r ) = (1 + 2 (r)) 21

(2.10)


Hen e, k(r) ! 0 as r # Æ and k(r) = O r at r = 1. We also have




r1+ 1 + o(1) at r = 1 : 1+ When  is equal to 0 or of type{I,  in reases from 0 to 1 as y in reases from Æ to 1. When  is of type{II, there exists Æ0 > Æ su h that (Æ0 ) = 0. So  de reases for y 2 (Æ; Æ0 ) and in reases to 1 in

 (r ) =

© 2001 by Chapman & Hall/CRC

32

CH. 2.

Invariant Solutions for the Curve Shortening Flow

(Æ 0 ; 1). The regularity of the spiral with its tip at the origin an be read

ds=dr

from (2.10). As

= (1 +

2 )1=2 ,

where

ds

is the ar -length

element, we see that the spiral is smooth at the tip if and only if 1= is a non-negative integer. When 1= is odd, we an extend it by setting


x(r); y(r)




x( r); y( r) ; r 2 (

=

1; 0) :

The resulting urve is a smooth, omplete, simple urve whose urvature is positive for all non-zero r . It is alled the \yin-yang" urve when

 = 1.

It is possible to mat h a type{I solution with a type{II

solution to form a smooth, omplete, embedded asymmetri spiral. To see this, let's hoose

+ and 

urves 

so that they blow up at the same

p

Æ. The orresponding start at (0; Æ ). We joint to

+ to form a omplete, embedded . We laim that it is smooth p at (0; Æ ). For, sin e d=dr > 0 on + and d=dr < 0 on near p (0; Æ ), we may represent as a fun tion y = y (), for  2 ( "; "), " small. When  2 ( "; 0℄, (resp.  2 [0; ")), is + (resp. ). To show that y is smooth at 0 we write (2.4) as h

L = (1 + L2 ) y 2 where

L

1

1 2 (1

+ L2 ) 2

1

1 2

1

L y

= 1= is regarded as a fun tion of

ontinuous in (

"; ")

and

L(0) = 0.

.

i

1

;

(2.11)

Noti e that

L

is

On the other hand, from (2.3)

we have

y = 2yL :

(2.12)

(2.11) and (2.12) together form a system of ODEs for y and L. Sin e the pair (y; L) is ontinuous in (

"; "), it is also smooth there.

In on lusion, we have proved the following results. positive

and point

X

in

R2 ,

there exists a spiral of the GCSF ro-

tating around the origin in onstant speed

© 2001 by Chapman & Hall/CRC

Given any

in the ounter lo kwise

2.3.

33

The support fun tion of a onvex urve

dire tion. Moreover, it has a unique in e tion point at X . The spiral is unique when X 6= (0; 0) and unique up to a rotation when X = (0; 0).

(a)

(b) Figure 2.1

Spirals for 2.3



= 1 : (a) the yin-yang urve and (b) an asymmetri spiral.

The support fun tion of a onvex urve

Before we dis uss self-similar solutions of the generalized urve shortening ow, we make a digression to review some basi fa ts of the support fun tion for a onvex urve. Let be a urve with positive urvature, i.e, it is uniformly

onvex. The normal angle1  at a point (p) is de ned by n(p) = ( os ; sin ). It is determined modulo 2. Just as the tangent angle, on e we x its value at a ertain point on the urve (p), the normal angles at other points on the urve are xed by ontinuity. The image of I under the normal map (p), J , is an open interval (1 ; 2 ) and is losed only if 2 1 is a multiple of 2. We may use the normal angle to parametrize the urve. In the following, we let = () so that n = ( os ; sin ). The support fun tion of

is a fun tion de ned in J given by h( ) = h ( ); ( os ; sin  )i: We 1 The Greek letter  is used to denote the tangent or the normal angle in dierent ontext. In the de nitions for the support fun tion, equations (3.9) and (4.1), it stands for the normal angle, while it means the tangent angles in other pla es.

© 2001 by Chapman & Hall/CRC

34

CH. 2.

Invariant Solutions for the Curve Shortening Flow

have

1 sin  + 2 os  :

h ( ) = Therefore,



an be re overed from

h

by

81 > < = h os  h sin ; > : 2 = h sin  + h os  :

(2.13)



The urvature of

an be expressed in terms of the support fun tion

in a very neat form. In fa t,

h + h

=

=

=

1 sin  + 2 os 



 d

ds ; t ds d

ds : d

By the Frenet formulas, we have

h + h =

1

k

:

(2.14)

The support fun tion an be des ribed as follows. the tangent line passing a point ( os ; sin  ). from the origin between

OP

P

Let

h

be

on the urve whose normal is

Then, the support fun tion is the signed distan e

O

to

`;

it is positive (resp. negative) if the angle

and ( os ; sin  ) is a ute (resp. obtuse). We note that

the support fun tion depends on the hoi e of the origin.

O

`

is hanged to

O, 0

When

the support fun tion is hanged from

hOO ; ( os ; sin )i:

h

to

0

The relationship between uniformly onvex, losed urves and their support fun tions is ontained in the following proposition whose proof is straightforward.

© 2001 by Chapman & Hall/CRC

2.4.

35

Self-similar solutions

Any 2n-periodi fun tion h with h + h > 0 determines a losed, uniformly onvex urve by (2.13) whose support fun tion is u. Two urves determined by h and h0 , respe tively, dier by a translation if and only if the dieren e of h and h0 is equal to C1 os  + C2 sin  for some C1 and C2 . Moreover, is embedded if and only if h is 2-periodi . Proposition 2.1

Let (
; t) be a family of uniformly onvex urves satisfying (1.2). We have p  e = p + F n ; t t

where e(; t) = (p(; t); t). Consequently, h(; t) = h e(; t); satis es ht =

F ( ;  +

ni



; k) ; (2.15) 2 where and k are given in (2.13) and (2.14), respe tively, and we have hosen the tangent angle to be  +=2. It is a paraboli equation if F is paraboli . For onvex ows, this is the most onvenient way to redu e (1.2) to a single equation. Noti e again by Proposition 1.1, (2.15) is equivalent to (1.2) as long as (
; t) is losed and uniformly

onvex. 2.4

Self-similar solutions

Now let's return to the GCSF. We seek solutions of this ow whose shapes hange homotheti ally during the evolution: b(
; t) = (t) (
). A

ording to Proposition 1.1, b is a self-similar solution if and only if its normal velo ity is equal to jb k j 1 b k . We have 0 
n = jk j

1

k:

When this urve is not at, 0  must be a non-zero onstant. After a res aling, we may simply assume the onstant is 1 or -1. When it is

© 2001 by Chapman & Hall/CRC

36

CH. 2.

Invariant Solutions for the Curve Shortening Flow

1

1, (t) = [(1+)t+ onst.℄ 1+ and so b expands as t in reases. When 1 it is -1, (t) = [ (1+)t+ onst.℄ 1+ , and so b ontra ts as to a point t in reases up to some xed time. We all the former an expanding self-similar solution and the latter a ontra ting self-similar solution. Noti ing that k n is independent of the orientation of the

urve, we may assume that k is positive somewhere. By introdu ing the normal angle and the support fun tion h, it is readily seen that h satis es the following ODE, 1 h + h = ; (expanding self-similar urve) (2.16) ( h)p or 1 (2.17) h + h = p ; ( ontra ting self-similar urve) ; h where p = 1=. We shall study (2.16) and (2.17) for  6 1. First, let's denote the solution of (2.16) subje t to the initial onditions h(0) = ;  > 0 and h (0) = 0 by h(; ). Clearly, for ea h  2 (0; 1), h(
; ) is an even, onvex fun tion whi h is stri tly in reasing in (0; ()) where () is the zero of h(
; ) and h (
; ) blows up as  " (). Further, we laim that h(
; 1 ) > h(
; 2 ) for 1 < 2 and () is stri tly in reasing from 0 to =2 as  goes from 0 to 1. To see this, we rst note that w = h(
; 1 ) h(
; 2 ) satis es w + (1 + q())w = 0 where q is negative. By the Sturm omparison theorem h(
; 1 ) > h(
; 2 ) in ( =2; =2). Next, the \normalized" support fun tion bh = h= satis es bh(0) = 1, bh (0) = 0 and it onverges to os  on every

ompa t subset of ( =2; =2) as  ! 1. Hen e, the laims hold. Consequently, for any  in (0; =2), the fun tion h( + 3=2; ), where  is given by  = (), determines a omplete expanding selfsimilar solution () lying inside the se tor f(x; y) : y < jxj tan g; it is the graph of a onvex fun tion u satisfying u(x)  (tan )x ! 0 and ux  tan  ! 0 as x ! 1. In summary, we have

© 2001 by Chapman & Hall/CRC

2.4.

37

Self-similar solutions

Proposition 2.2 Every expanding self-similar solution is ontained in a omplete expanding self-similar solution, whi h is ongruent to exa tly one

() after the multipli ation

of a onstant.

Next, denote the solution of (2.17) subje t to the initial onditions h(0) =  > 1, h (0) = 0, by h(; ). We note that it admits a rst integral 1 2 2 h1 p (h + h ) ; (p 6= 1); 2 1 p or 1 2 2 (h + h ) log h; (p = 1): 2  It follows that, for ea h , the fun tion h(
; ) is periodi in . Its maximum M = h(0) and minimum m are related by 1 2 (M 2

m2 ) =

1 (M 1 1 p

p

m1 p);

(p 6= 1)

or

M 1 2 (M m2 ) = log ; (p = 1): 2 m Hen e, when p > 1, m ! 0 as M =  ! 1. Also, it is lear that h is on ave on fh > 1g and onvex on fh < 1g. We laim that, p p for any T 2 (; 2= p + 1)(p < 3) or (2= p + 1; )(p > 3), there exists some  > 1 su h that h(
; ) has period T . To see this, we rst observe that h(
; 1)  1 is a trivial solution, whi h orresponds to the unit ir le. The linearized equation of (2.17) at this trivial solution is given by 00 + (1 + p) = 0; 0 (0) = 0 :

p

It admits ' = os p + 1  as its solution. Therefore, for small ", p h(; 1+ ") = 1+ " os p + 1  + O("2 ), and so the period of h(
; 1+ ") p is very lose to 2= p + 1. On the other hand, it is not hard to see that the normalized support fun tion h= tends to j os j uniformly

© 2001 by Chapman & Hall/CRC

38

CH. 2.

Invariant Solutions for the Curve Shortening Flow

Rnf=2 + n; n 2 Zg. By ontinuity, we p

on lude that every number between  and 2= p + 1 is a period of h. When p = 3, all solutions of (2.17) are given by

on every ompa t set in

2 os2 ( 0 ) + 

(

2

2 sin (

 0 ))1=2 ;  2 [1; 1); 0 2 [0; ) :

They are support fun tions of ellipses.

Figure 2.2 Contra ting self-similar 3-petal urves for

 = 0:7; 1; 1:2.

We shall use the following hara terization of the ir le in the next hapter.

© 2001 by Chapman & Hall/CRC

2.4.

39

Self-similar solutions

Proposition 2.3 For

 > 1=3,

the only losed, embedded ontra t-

ing self-similar solutions are ir les.

h = h(; ) be a 2-periodi solution of (2.17) satisfying h(0) =  > 1 and h (0) = 0. We have

Proof: Let



h

(  ) +

1+

hp+1

By the Sturm omparison theorem,

; ).

(0

point.



p

h

h = 0:

must vanish at some point in

h is symmetri with So the minimal period of h, T , is at most  . But then it means that

respe t to the

Next, we have 1 2



2 2 (h ) + 4(h )



(3

=

p)h : h

Therefore, 2(3

p)

Z

Z T

T h

os 2( 0 )[(h2 ) + 4(h2 ) ℄

os 2( 0 )d = h 0 0 T

 0 )(h2 ) :

= os 2(

0

If we hoose 2 be omes

T

= (



=2),

the right hand side of this identity

h2 (0)

1

0



(0)

os 20 ; hp 1 whi h is positive be ause T 6  . On the other hand, os 2(T=2 0 ) = os =2 = 0 and h < 0 (resp. > 0) in (0; T=2) (resp. (T=2; T )). 4

Therefore, its left-hand side is negative. The ontradi tion holds. We



have shown that the only 2 -periodi solution of (2.17) is

h(
; 1).



In the rest of this se tion, we examine losed ontra ting selfsimilar solutions for the CSF more losely. They are sometimes alled Abres h-Langer urves. It was shown in Abres h-Langer [1℄ that the period of

© 2001 by Chapman & Hall/CRC

k, T (),

40

CH. 2.

Invariant Solutions for the Curve Shortening Flow

p stri tly de reases from 2 to  as  in reases from 1 to 1. Consequently, for any pair of relatively prime natural numbers m and n, p satisfying m=n 2 (1; 2), there orresponds a losed Abres h-Langer

urve whi h rotates around the origin m times and has n petals. p Moreover, no other values in (1; 2) yield losed urves. To re on ile the notations here with those in [1℄, we note that the rst integral of (2.17) (p = 1) is 1 2 1 2 1 2 h + 2 h = log h +  + 2 ; where  = 12 2 12 log  satis es (1) = 0, i.e., it vanishes at the unit ir le. On the other hand, letting B = 2 log k and observing that k = h, it follows from (2.17) that Bss + 2(eB

and

1) = 0 ;

1 B 2 + 2(eB B 1) = 2 : 2 s They are pre isely equations (2) and (3) in [1℄ where we have taken  to 1. The total hange in tangentnnormal angle within a period of k is given by Z Bmax 2dB p () = B 4e [ (eB B 1)℄ Bmin hZ max (2.18) 2dh = p 2 2 (h 2 log h 1) hmin = T () : Proposition 3.2 (v) in [1℄ asserts that () is stri tly de reasing in , and so T is stri tly de reasing in . Next, we onsider the linearized stability of the Abres h-Langer

urve. Let wm;n be the support fun tion of the Abres h-Langer urve

© 2001 by Chapman & Hall/CRC

2.4.

41

Self-similar solutions

orresponding to (m; n). Consider the eigenvalue problem with periodi boundary ondition in the spa e Sm onsisting of all periodi fun tions in [0; 2m℄,

L' = ' +

1+

 = 2' ; w

1


w2

'

(2.19)

where w = wm;n . It is well-known that (2.19) has in nitely many eigenvalues satisfying  0 < 1

6 2 < 3 6 4 <


! 1 ;

and the orresponding eigenfun tions '2j 1 and '2j have exa tly 2j zeroes in Sm . One an verify that '0 = w and 0 = 2. Also, the fun tions os  and sin  are eigenfun tions for the eigenvalue 1. Proposition 2.4

When w

of (2.19) and 2n 1

= 0.

6 1, w is the (2n

1) st eigenfun tion

Moreover, zero is a simple eigenvalue.

Sin e w 6 1, w is nontrivial and satis es (2.19) for  = 0. As it has exa tly 2n zeroes, it is either the (2n 1) st or the 2n th eigenfun tion. To show that it is the former, we onsider the   Neumann eigenvalue problem in 0; m=n :

Proof:

8 N > > > < ' = w2 ' ;

L

> > > :' (0)

= '

m
=0: n

(2.20)

Sin e w is even, ea h eigenfun tion of (2.20) an be extended by odd fun tion and then periodi ally to an eigenfun tion of (2.19) with the same eigenvalue. In parti ular, the eigenfun tion '2 orresponding to 2N has exa tly 2n zeroes in Sm . As w is the lowest Diri hlet   N eigenvalue of L in 0; m=2 and it is well-known that D 1 > 2 , w

© 2001 by Chapman & Hall/CRC

42

CH. 2.

Invariant Solutions for the Curve Shortening Flow

and '2 are, respe tively, the (2n 1) st and 2n th eigenfun tions of (2.19). To show that zero is simple, we re all that w() = h(; 0 ) for some 0 > 1. So u() = h(; )= =0 is also a solution of (2.19) with  = 0. However, sin e the period stri tly de reases in , u

annot belong to Sm . So zero must be a simple eigenvalue.  It follows that, when w 6 1, the stable invariant manifold of w is of odimension 2n and the unstable invariant manifold has dimension 2n 1. When w  1, the eigenfun tions of (2.19) an be found expli itly. It is straightforward to verify that the stable invariant manifold of w is of odimension N+ and the unstable invariant manifold has dimension N+ where 

N+ = q 2 Z : 2

q
2 >0 : m

Noti e that now zero is no longer an eigenvalue of (2.19).

Notes The grim reapers were used in Grayson [66℄, Alts huler [3℄, and Hamilton [72℄. It an be hara terized as the eternal solution (a solution exists for all t 2 R) with non-negative urvature. Its stability was studied in Alts huler-Wu [5℄. In the ontext of the CSF, the spiral was rst pointed out in Alts huler [3℄, where it is alled the \Yin-Yang urve." However, spirals arising from the urvatureeikonal ow have been studied extensively in mathemati al biology and hemi al rea tion. See, for instan e, Keener-Sneyd [84℄, Murray [90℄, and Ikota-Ishimura-Yamagu hi [80℄. The expanding self-similar solution for the mean urvature ow was known ba k to Brakke [22℄, who alled it \evolution from a orner." See E ker-Huisken [46℄ for

© 2001 by Chapman & Hall/CRC

2.4.

Self-similar solutions

43

a result on erning its stability. The Abres h-Langer urves were

lassi ed in [1℄ and further results on their stability an be found in this paper and Epstein-Weinstein [49℄. Proposition 2.4 is taken from [49℄. Our dis ussion on the linearized stability slightly diers from [49℄ be ause they use the equation for k = k(s) instead of (2.17). Noti e that, in their formulation, the eigenfun tions os  and sin , whi h orrespond to translating the urve in the plane, do not ome up. Finally, we mention that a ertain ontra ting spiral was used in the study of the formation of singularities in Angenent [15℄ and Angenent-Velazquez [18℄. All spe ial solutions dis ussed in this hapter are group invariant solutions. A systemati investigation of the group invariant solutions for the GCSF an be found in Chou-Li [31℄. Using Lie's theory of symmetries for dierential equations, they determine the symmetry group and obtain the optimal system (the largest family onsisting of mutually nonequivalent group invariant solutions) for the GCSF. Many new spe ial solutions are found for the aÆne CSF. In this

onne tion, also see Calabi-Olver-Tannenbaum [25℄.

© 2001 by Chapman & Hall/CRC

Chapter 3 The Curvature-Eikonal Flow for Convex Curves In this hapter, we shall study the anisotropi urvature-eikonal ow (ACEF) for losed onvex urves. Three ases| ontra ting to a point, onverging to a stationary shape, and expanding to in nity{ will be studied in detail in Se tions 4, 5 and 6. In Se tion 3, we present a self- ontained treatment of an important spe ial ase| the urve shortening ow. Some suÆ ient onditions whi h ensure shrinking to a point are established in Se tion 2. They apply to ows more general than the ACEF and will be used in later hapters. 3.1

Blas hke Sele tion Theorem

The following basi ompa tness result for onvex sets is very basi .

Let fKj g be a sequen e of onvex sets whi h are ontained in a bounded set. Then there exists a subsequen e fKjk g and a onvex set K su h that Kjk

onverges to K in the Hausdor metri .

Theorem 3.1 (Blas hke Sele tion Theorem)

Re all that the Hausdor metri for two arbitrary sets A and B are given by 45 © 2001 by Chapman & Hall/CRC

46

CH. 3.

d(A; B )

The Curvature-Eikonal Flow for Convex Curves

 = inf  > 0 : A  B + D1 ; B  A + D1 ;

where D1 is the unit disk entered at the origin. The onvergen e in Hausdor metri for onvex sets is related to the uniform onvergen e of their support fun tions. In fa t, for any

onvex set K , its support fun tion is given by n o h( ) = sup h(x; y ); ( os ; sin  )i : (x; y ) 2 K : When the boundary of K , K , is uniformly onvex, this de nition is the same as the one given in x2.3. We extend h to be ome a fun tion of homogeneous degree one in the whole plane by setting H (x; y )

= rh() ;

(x; y) = (r os ; r sin ) :

Then H is onvex. To see this, we note that, when K is uniformly

onvex, the Hessian matrix of H is given by 3 2 sin2  sin  os  7 h + h 6 7 ; 6 5 4 r 2 sin  os 

os  and it is non-negative de nite. The general ase then follows by approximation. There is a one-to-one orresponden e between bounded onvex sets and onvex fun tions of homogeneous degree one in the plane. A tually, any su h fun tion H determines a onvex set de ned by n o K = (x; y ) : h(x; y ); ( os ; sin  )i 6 h( ); for all  2 [0; 2 ) : It is not hard to he k that the support fun tion of K is h. Now, it is lear that fKj g onverges to K in the Hausdor metri

© 2001 by Chapman & Hall/CRC

3.2.

47

Preserving onvexity and shrinking to a point

if and only if fHj g onverges to H uniformly in every ompa t set. We an see why the Blas hke Sele tion Theorem holds using this

onne tion. First of all, by onvexity, sup rH C1

sup jH j

6 dist C(2C ; C ) 1

2

for any bounded open sets C1 and C2 with C1  C2 . When fKj g is

on ned to a bounded set, fHj g is uniformly bounded. Hen e, we

an sele t a subsequen e fHjk g whi h onverges uniformly to some

onvex fun tion H . In other words, fKjk g onverges to the onvex set determined by H in the Hausdor metri . 3.2

Preserving onvexity and shrinking to a point

We rst show that onvexity is preserved for a large lass of (1.2). Proposition 3.2 (preserving onvexity) Let be a maximal so
2 1

lution of (1.2) in for positive

t

C S



[0; !)

. Then,

(
; t)

is uniformly onvex

under either one of the following onditions:

R2  S 1  (0; 1) and 0 is uniformly onvex ;

(a)

F

is paraboli in

(b)

F

is paraboli in

 2 F=y 2

=0

at

R2



q = 0,

S1



and

[0; 1),  2 F=x2 =  2 F=xy

0

=

is onvex.

We rst prove the proposition under (a). Introdu e the support fun tion as long as (
; t) is uniformly onvex. By (2.14) and (2.15), the urvature k = k(; t) satis es Proof:

 2F  k = k2 + F : t  2

We may rewrite the equation as kt = k 2 Fq k + Gk ;

© 2001 by Chapman & Hall/CRC

(3.1)

48

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

where G is bounded in S 1 v = et k satis es

 [0; T ℄ for any T

< ! . The fun tion

vt = k 2 Fq v + (G + )v :

 We hoose  so that G+ is non-negative. Let vmin(t) = min v(; t) :  2 [0; 2 ) . It is not hard to show that vmin in Lips hitz ontinuous and dvmin(t)=dt > v=t(; t) at vmin (t) = v(; t). So, dvmin dt

> (G + )vmin > 0 :

In other words, k(; t) > e t mink(; 0) > 0.  Next, assume (b) holds. It suÆ es to show that k be omes positive instantly. Let's look at the lo al graph representation of the

ow. By dierentiating (1.3) and using (b) k = k(x; t) satis es a 1 paraboli equation of the form kt = (1 + u2x ) 2 Fq kxx + bkx + k, where b and are bounded. It follows from the strong maximum prin iple that k is positive for t > 0.  Let (
; t) be a uniformly onvex solution of (1.2) where F is paraboli in R2  S 1  (0; 1). Suppose further that F satis es (i) there exists a onstant C0 su h that F (x; y; ; 0) > C0 in R2  S 1 , (ii) for ea h (x; y; ), lim F (x; y; ; q) = 1, and (iii) for Proposition 3.3

any ompa t set K in R

2



S1,

q

!1

there exists a onstant C1 su h that

jF j + jFxj + jFy j 6 C1 (1 + qFq ) in R2  S 1  [0; 1). Then, if ! is nite, the area en losed by (
; t) tends to zero as t " !.

First of all, by paraboli ity, assumption (i) and Proposition 1.6, (
; t) are on ned to a disk of radius diam 0 + C0 !. Suppose on the ontrary that there exists a sequen e ftj g, tj " ! su h that the area en losed by (
; tj ) does not tend to zero. By the Blas hke Proof:

© 2001 by Chapman & Hall/CRC

3.2.

49

Preserving onvexity and shrinking to a point

Sele tion Theorem, we may sele t a subsequen e, still denoted by f (
; tj )g, su h that ea h (
; tj ) ontains a disk Dj , and fDj g on
verges to the disk D4 (x0 ; y0 ) for some (x0 ; y0 ) and . By applying the strong separation prin iple to (
; t) and the ow starting at

D4 (x0 ; y0 ) , we know that D2 (x0 ; y0 ) is en losed by (
; t) for all t suÆ iently lose to !. Use (x0 ; y0 ) as the origin and introdu e the support fun tion h(; t). Then h(; t) > 2 on [t0 ; ! ) where t0 is lose to ! . We laim that the urvature is uniformly bounded in [t0 ; !). To prove this, we

onsider the fun tion = h

ht 

= hF 

:





Let (0 ; t1 ) = max (; t) : (; t) 2 S 1  [t0 ; T ℄ where T < !. In view of (ii), we may assume (0 ; t1 ) is positive. If t1 > t0 , we have 0 =  = h ht + (hht h )2 2

0 6 t = h htt + (h ht )2

(3.2)

; ;

and

(3.3)

2ht h2 ; (3.4) (h )3 at (0 ; t1 ). On the other hand, from (2.15) and (3.1), we have 0 >  =

htt

=

ht h 

Fx

+ (2hht h)2 + (hht h)2

 1 t

Fy

 2 t

Fq k 2

 2F

 2

+F



:

Therefore, by (3.3), (3.2), and (3.4), 0 6

Fx

 1 t

Fy

 2 t

Fq k 2

ht + F




6

Fx

 1 t

Fy

 2 t

+ Fh Fq k

F Fq k 2 h 

+ hF

© 2001 by Chapman & Hall/CRC

+ hF

2



2



:

50

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

By (2.13), we have Fx

 1 t

2

+ Fy  = t

F Fx os  + Fy sin 




+ F (Fx sin h)

Fy ( os  )h : 

By onvexity, we know that jh j 6  sup jhj. Therefore, it follows from (iii) that (0 ; t1 ) is bounded above. But then (ii) implies that the urvature is uniformly bounded in [t0 ; !), whi h is impossible by Proposition 1.2. Hen e, the en losed area must tend to zero.  Let (
; t) be a uniformly

onvex solution of (1.2) where F is paraboli in R2  S 1  (0; 1). Suppose that ! is nite and the en losed area tends to zero as t " !. Then, (
; t) shrinks to a point under either one of the following

onditions: (i) F (x; y; ; 0) = 0, or Proposition 3.4 (shrinking to a point)

(ii) F is uniformly paraboli in R2 only depends on . Proof:

 S 1  [0; !) and F (x; y; ; 0)

By (3.1), we have F t

= k2 Fq

 2F

 2

+F



:

So dFmin dt

> k2 Fq Fmin :

In ase the ow does not tend to a point as the en losed area approa hes to zero, learly, kmin(t) tends to zero. A

ording to (i), Fmin (t) tends to zero too. But, the inequality above shows that Fmin has a positive lower bound. So the ow must shrink to a point. Next, assume (ii) is valid. By uniform paraboli ity, F 6 C (1+k). By omparing the ow with a large expanding ir le with normal velo ity C (1 + k), we know that is ontained in a bounded set. If

© 2001 by Chapman & Hall/CRC

3.3.

51

Gage-Hamilton Theorem

 it does not shrink to a point, we an nd (
; tj ) , tj " !, su h that

(
; t) onverges to a line segment. For simpli ity we may take this  segment to be (x; 0) : jxj 6 `=2 . By (ii), verti al lines translate in

onstant speed under the ow. By the Sturm Os illation Theorem, for all t suÆ iently lose to !, (
; t) over ( `=4; `=4) is the union of the graphs of two fun tions U1 and U2 , U1 (x; t) < U2 (x; t), whi h

onverge to 0 together with bounded gradients. Now we an follow the argument in the proof of Proposition 1.9 that the urvature of

over [ `=8; `=8℄ is uniformly bounded near ! . However, then by the strong maximum prin iple, the urvature annot be ome zero at t = ! . The ontradi tion holds.  3.3

Gage-Hamilton Theorem

The main result of the urve shortening ow for onvex urves is

ontained in the following two theorems of Gage and Hamilton.

Consider the Cau hy problem for the urve shortening ow, where 0 is a onvex, embedded losed urve. It has a unique solution (
; t) whi h is analyti and uniformly onvex for ea h t in (0; !) where ! = A0 =2 and A0 is the area en losed by 0 . As t " !,

(
; t) shrinks to a point. Theorem 3.5

This theorem follows from Proposition 1.1, 3.2{3.4, and (1.17). To study the longtime behavior of the ow, we res ale the urve by setting

e (
; t) =

   12 

(
; t) A (t )



(
; ! ) ;

where A(t) = A0 2t. Then the area en losed by this normalized

urve e is always equal to . The normalized support fun tion and

© 2001 by Chapman & Hall/CRC

52

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

urvature are given by


=

e h ( ; t)

  1=2

A (t )

h(
; t) ;

and


=

e k ( ; t)

 A(t) 1=2



k (
; t) ;

respe tively. They satisfy the equations =

e h

e k+e h;

(3.5)

and e k

= ek2

e k




+ ek

(3.6)

e k;

where 2 = log(A0 2t). The normalized ow e(
;  ) is de ned in [0 ; 1) for 0 = 2 1 log A0 . Theorem 3.6

e

onverges to the unit ir le entered at the origin

smoothly and exponentially as



!1

.

The proof of this theorem will be a

omplished in several steps. First we have the following gradient estimate. Lemma 3.7

S

sup 1

[0 ; ℄

e k2 + e k2 




6 max

n S

sup 1

[0 ; ℄

e k2 ;

S




sup 1

f0 g

e k2 + e k2 


o

:

Consider the fun tion  = ek2 +ek2 . Suppose that (0 ; 1 ) = sup(k2 + k2 ) and 1 > 0 . We laim that ek must vanish at (0 ; 1 ).
For, if not,  = 2ek ek + ek = 0. We must have ek + ek = 0. Using Proof:

© 2001 by Chapman & Hall/CRC

3.3.

53

Gage-Hamilton Theorem

 6 0 and  > 0 at (0 ; 1 ), we have 0 6

e ke k

+ ek ek

6

e k2 + e k e k2 e k

6

e k2

+ ke




e k2 

e k2 ;

whi h is impossible. So ek (0 ; 1 ) = 0.  Suppose that ekmax ( ) = ek(0 ;  ) and ekmax ( ) > ek(;  ) for all  2 [0 ;  ℄. It follows from this lemma that 0

0

e kmax ( )

e k (;  )

6 j

0 j sup je k (
;  )j

6 j

0 j C0 + e kmax ( ) ;






where C0 only depends on the initial urve. Therefore, for  satisfying j 0j 6 1=2, we have e kmax ( )

6 2ek(;  ) + C0 :

(3.7)

Next, we derive an upper bound for the normalized urvature. The key step is the monotoni ity of the entropy for the normalized

ow. For any uniformly onvex urve , its entropy is de ned by Z 1 E ( ) = 2 k log kds

=

Z

log kd :

Lemma 3.8


d E

e(
;  ) 6 0 : d

© 2001 by Chapman & Hall/CRC

54

CH. 3.

Proof:

The Curvature-Eikonal Flow for Convex Curves

By (3.6), we have dE = d

Z e k ; ke

and d2 E d 2

 Z ek2 Z  = 2 + e2 k dE  dE

>

2

d d

ek  ke 

+1 :

Were dE =d positive somewhere, it blows up at some later time. Sin e now E ( ) is de ned for all  > 0 , this is not possible. So dE =d 6 0 everywhere.  The width of a onvex urve in the dire tion ( os ; sin ) is given by w() = h() + h( + ). We show that an upper bound on the entropy yields a positive lower bound for the width. Lemma 3.9 Let

be a losed, onvex urve and

tion. There exists an absolute onstant

C0

h

its support fun -

su h that

w( ) > C0 e E ( ) for all

.

Proof:

For any xed 0 ,

Z +0 0

sin(

k

0 )

d =

Z +0

sin(

0

= w(0 ) ;

© 2001 by Chapman & Hall/CRC

0 )(h + h)d

3.3.

55

Gage-Hamilton Theorem

after performing integration by parts. By Jensen's inequality, log w(0 ) = log

>

1



1



Z

Z+

 0

0

j sin(

k ( )

0 )j

d + log 



0

1

log sin d + log 

Z



0

log k( + 0 )d :

A similar inequality an be obtained when we integrate from  + 0 to 2 + 0 . Hen e, the lemma follows.  It follows from Lemmas 3.8 and 3.9 that the width of e(
; t) in all dire tions are uniformly bounded from zero. Sin e the area en losed by e is always equal to , the length and diameter of e are bounded from above and its inradius has a positive lower bound. Now we pro eed to bound the urvature of e

. If e k is unbounded, we an nd e e fj g, j ! 1, su h that kmax(j ) > kmax( ), for all  6 j . Then for ea h  = j , we an use (3.7) to get 2E (0)

> > >

Z

log e k (;  )d

Z

j

0

Z

log e k (;  )d +

j6 12

ek log ekds

fek ekmin

e k2

min




1

;

e kmin < 1 ;

!1=2

58

CH. 3.

we have

q

log

1

2 ekmin ( )

ekmin( )

The Curvature-Eikonal Flow for Convex Curves

q

6 log

1

2 kmin (j )

ekmin(j )

+ j +1

j :

Using sj +1 sj 6 2 and ekmin(j ) tends to 1 as j ! 1, we on lude that e kmin ( ) has a positive lower bound. With two-sided bounds on the urvature, we an use standard paraboli regularity to obtain all higher order bounds. It follows from (3.8) that lim



!0

Z e (h

e)2 k (;  )d = 0 ehek

So eh(
;  ) onverges to 1 smoothly. To nish the proof, we show the onvergen e is exponential. Lemma 3.11 For any

Z

Proof:

" > 0,

2 e > (4 k

there exists

")

Z

ek2

;

for

1

>



>

0 su h that

1

:

As (1=e k ) is orthogonal to 1, sin  and os  , Z  2 Z  2 1 >4 1 :

ek

ek





The desired inequality follows from the fa t that e k tends to 1 smoothly.



By Lemma 3.11, for large  , we have 1 d 2 d

Z

ek2 =

6 Therefore,

© 2001 by Chapman & Hall/CRC

Z

ek2 6 e2(2

Z

(2

Z 2 ek2 ek + (3e k2 ")

Z

")(1  )

Z

ek2

:

ek2(
; 1 ) :

1)e k2

3.4.

59

The ontra ting ase of the ACEF

For ea h  , we an nd 0 su h that ek(0 ;  ) = 1. So, for any " > 0, there exists C" and " su h that e k



(;  ) 1 6 C" e

3.4

")

(2

;

all



> " :

The ontra ting ase of the ACEF

In this se tion, we begin the study of the ACEF, whi h is now pla ed in the following form,  t





= ()k +  () n ;

(3.9)

where  and are positive 2-periodi fun tions of the normal angle and  2 R. When   1,  1, and 0 is a ir le of radius R, the ow (3.9) shrinks to a point (resp. expands to in nity) when 1=R +  > 0 (resp. 1=R +  < 0), and it is stationary when 1=R +  = 0. It turns out this behaviour is typi al. In fa t, by Propositions 3.2{3.4, we know that any solution of (3.9) starting at a losed, onvex urve is uniformly onvex for ea h t, and ! is nite if and only if it shrinks to a point. Using the strong separation prin iple, we also know the following fa ts hold: (a) Let i be solution of (3.9) , i = 1; 2, satisfying 1 (
; 0) =

2 (
; 0), and 1 < 2 . Then, 2 (
; t) is en losed by 1 (
; t) as long as 2 (
; t) exists. i

(b) If (
; t) is en losed in a ir le of radius R, 1=R >  max=min, at some t, it shrinks to a point; if (
; t) en loses a ir le of radius R, > max=( min), it expands to in nity. Let 0 be a xed losed, onvex urve and (
; t) the solution of (3.9) starting at 0 . Let n

A =  : (
; t)

© 2001 by Chapman & Hall/CRC

o

shrinks to a point

;

60

CH. 3.

and B

=

n

 : (
; t)

The Curvature-Eikonal Flow for Convex Curves

o

expands to in nity

:

From the above fa ts, we know that A and B are open intervals of the form ( ; 1) and ( 1;  ) where  >  and  < 0. Now we

an state our main result. Theorem 3.12 Consider (3.9) , where  and are positive and

2 R.

For any losed, onvex 0 , there exists  < 0 su h that the Cau hy problem for (3.9) has a unique solution (
; t) whi h is uniformly onvex for ea h t in (0; !). Moreover,  =  and the following statements hold: (i) When  >  , ! is nite and (
; t) ontra ts to a point as t " ! . Moreover, if we normalize (
; t) so that its en losed area is onstant, the normalized ow sub onverges to self-similar solutions of the ow 

 t

= ()kn :

(3.10)

(ii) When  =  , ! = 1 and the urvature of (
; t) is uniformly bounded. The ow onverges smoothly to a stationary solution of (3.9) if and only if there exists a 2-periodi fun tion  su h that

 = d2  +  : d2 (iii) When  <  , ! = 1 and the ow expands to in nity as t ! 1. If the polar diagram of 1= is uniformly onvex, i.e., + d2 =d2 > 0 for all , then (
; t)=t onverges smoothly to the boundary of the Wul region of  . The Wul region of is given by f(x; y); h(x; y); ( os ; sin )i 6 ()g.

© 2001 by Chapman & Hall/CRC

3.4.

61

The ontra ting ase of the ACEF

In the rest of this se tion we prove Theorem 3.12 (i). Let F = k +  . Then F satis es 

1 F

t

= k2

F

+F




(3.11)

:

We begin with a gradient estimate for F . Lemma 3.13 Either

F (; t) t

>0

F 2 + F2




or

where

M2

n

= max sup

F 2 + F2

; t





6 M2 ;

o
; 0 ; sup .





For any xed (0 ; t0 ), t0 > 0, let B = F 2 + F2 1=2 0 ; t0 . Suppose that B > M . We are going to show that (F + F ) > 0 at (0 ; t0 ). Choose  2 ( ; ) su h that

Proof:

F ( 0 ; t 0 )

= B os  ;

F (0 ; t0 ) = B sin  : 

Consider the fun tion G = F F  , where F  = B os( 0 +  ) is a stationary solution of (3.11). By assumption, G is positive at (0   ; t), 0 6 t 6 t0 , and has a double zero at (0 ; t0 ). Sin e B > M , G(; 0) must vanish somewhere in (0  ; 0  +  ).

© 2001 by Chapman & Hall/CRC

62

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

Suppose that 1 is a root in (0 G (1 ; 0) = 



 ). We have

; 0

F (1 ; 0) + B sin(1  

6




0 for any root 2 in (0 ; 0  + ). Therefore, G(
; 0) has exa tly two roots in (0  ; 0  +  ). By the Sturm os illation theorem, G(
; t0 ) has no roots other than 0 . So F + F = G > 0 at (0 ; t0 ).  We an dedu e a gradient estimate from this lemma. In fa t, let's assume (F + F )(1 ; t1 ) > 0 and F (1 ; t1 ) > 0. We an nd an interval (1 ; 2 ) on whi h F +F is nonnegative and F (2 ; t1 ) 6 M . Therefore, we have F (1 ; t1 )

6 F (2; t1 ) +

Z  2

1

F d ;

whi h implies sup 

F

(; t)



6M+

2 Z F

(; t)

d

:

(3.12)

0

It follows immediately that Fmax (t)

6 6

© 2001 by Chapman & Hall/CRC

Z












F (; t)d + 2 sup F ( ; t) 

M1 1 +

Z 2 F

0

(; t) d ;

(3.13)

3.4.

63

The ontra ting ase of the ACEF

and Fmax (t) 6 2F (; t) +

M 2

(3.14)

for all , j 0 j 6 1=(4) where Fmax (t) = F (0 ; t). Now we study the normalized ow. First, from (1.19), we have A(t) lim t"! 2(! t)

Z 2 1 ()d : =2 0

So we let

e(
; t) =


1=2

2! 2t

(
; t) :

R The area en losed by e approa hes the onstant 21 02 d as t " !. Changing the time s ale from t to 2 = log(1 ! 1 t), the equations for the normalized support fun tion and the normalized urvature are given, respe tively, by 

e h 

=

e k 

= ek2

p

ek + 2! e

  2 (k )

 2









+ ek

+ eh ;

e k+

p

2! e

(3.15) 

e k2




 + : (3.16)

The entropy for the normalized ow is de ned by


Z

E e(
;  ) = () log ek(;  )d : We shall show that it is uniformly bounded for all  2 [0; 1).

© 2001 by Chapman & Hall/CRC

(3.17)

64

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

We have d d

= = =

Z

Ee( ) ()  ek (;  )d ek (;  ) 

2 0

Z

2

(

0

Z

2

0

(3.18)

ek

 2 (e k) 2 

u(;  )d +

Z

! + ek

2

)

p

 + 2!e  ek (  + )

hp

d

i

2!e  ek (  + ) +2e  ek

0

d ;

where u = e k

 2 (e k) 2 

! + ke

 2e  ek ;

and  > 0 to be spe i ed later. Let's ompute d d

= =

d d

Z

Z Z

2

0

2 0 2 0

(

ud

2  2 4 ek

  e 2 k ek 

)

2e ek  

© 2001 by Chapman & Hall/CRC

d

 (e k) 

2

!2

3

 2e  ek5 d

 (e k) 

!

 (e k ) 

!

+ 2e  ek

3.4.

65

The ontra ting ase of the ACEF

Z 2 "

! # + ek ek + e  ek e  ek  d = 2 0 ! " ! Z 2  2 (ek) 2 (e  k ) = 2 + ek  ek2 2 + ek ek  2 0 # Z 2 p  e2 2e  ekd + 2!e k  (  + ) d + 0 " ! Z 2 2 e  ( k ) 2  e e  k 2 + ke ke  2 0 # p  e2 + 2!e k  (  + ) d !#2 ! Z 2 8 < 1 "  2 (ek) 2 (e  k ) ke 2 + ke = 2 ke 2 + ke 0 : ! ) p  e2  2(ek) e e

 2 (k )  2

+ 2!e k +2

+2

Z 2 0

Z 2

e



e



 2

ekd 2 ekd 2

+ k

Z 2 0

e

Z 2 p

 ( 

+ )

ek

 2 (k )  2



2!e

2

2

d

e

! + ek d

ek2  (  + ) d

0 0 !2 ! Z 2 1 8 < 2  2(ek) 2 (e  k ) e + ek = 2 2ek 2 + ek   :(k )  2 0 ! ! 2 (e 2 (k e  k )  )  2 + ke  4e (ek) + ek +ke  2  2 ! 2 (e  k )  2 +4e (ke) + ke + 4e  ke2 4e  ke2  2 o +2 2 + 42 e 2 (ek)2 42 e 2 (ek )2 d ! Z 2 p  2 (e k)  e2 + ek  (  + ) d +2 2!e k 2 0

© 2001 by Chapman & Hall/CRC



66

CH. 3.

+2

Z 2 0

Z 2



e

The Curvature-Eikonal Flow for Convex Curves

ekd 2

Z 2 0



e

ek2

p Z 2

ekd 2 2!

e

 2 (k )  2

! + ek d

e + ) d ! Z 2 1 Z 2 2 (e  k ) 2 e e = 2  u d + 2 0 k 2 + k d + 0 ! Z 2 Z 2 2 (e  k )  e2 e e k 8 e  e k d + k d 8  2 0 0 Z 2 Z 2 e2d + d 82 e 2 k 2 0 ! Z 20 p 2 (e  k )  e2 e +2

2

0

+4

0

2!e k

Z 2 0



e

Z 2 1

0

u

0

82

Z 2

+2 2!

0

2 2!

= 2

0

e

+4 +4 2

Z 2 0

Z 2

Z 2 0 0

0

u

e

e

Z 2 0

2



ud + 6

ek

e

e 2 k 2  ( 

d + 2

e

"



e

Z 2 0

ek ek

ud

e

 2 (k )  2

 ( 

ek2

+ ) d +

e

 2 (k )  2

! + ek d

+ ) d

 2 (k )  2

2



e

e

p Z 2

Z 2 1

0

e 2 k 2 d

p Z 2 0

Z 2

e 2 k 2  ( 

d + 2

2

0

e 2 k 2  ( 

+ k

 2

ekd 2

p Z 2

2 2!

= 2



e

Z 2 0

e



ek2

e

 2 (k )  2

! + ek d

! + ek  (  + ) d

+ ) d

! # + ek  2e  ek d

Z 2 2 e e k d + 8 e 2 e k 2 d + 0 !h i 2 (e p  k )  e2 e k +  k  + 2 ! ( + ) d   2

© 2001 by Chapman & Hall/CRC



3.4.

67

The ontra ting ase of the ACEF

p Z 2 2 e2 2 e e k  (  + ) d 8 e k d 2 2! 0 0 Z 2 Z 2 Z 2 1  e e 2 = 2  u d + 2 0 (1 + 2e ! k)u d + 4 0 e kd 0 Z 2 i e) e h p  2 (k  e2 2

+2

Z 2

k

e

0

p Z 2

2 2!

0

Using 2

Z 2 0

2

e

e 2 k 2  ( 

2e ekud = 2 

> we get d d

Z 2 0



0

+4

Z 2

u d +

Z 2

2e  1=2 ek u

2



e

 + 2! (  + )

d

+ ) d:

Z 2 1 0

0

Z 2 1

>

ud

+ k

 2

2

Z 2 0

4

d

1=2

Z 2

2 e

0

2ud 4

2

ekd + 2

Z 2

e

ud

Z 2 0



2

ek2 d;

e

e 2 k 2 d

ek2

! i p + ek [+ + 2! (  + ) d p Z 2 2 e2 2 2! e k  (  + ) d: 0

e

 2 (k )  2

0

Now, hoosing

p

 = 2! max jj j  + j 

we have

Z 2

d d

>

0

Z 2 1

ud 2

u

Z 2

0

+ 4 + 2

0

Z 0

0 2

e

d +

e 

© 2001 by Chapman & Hall/CRC



Z 2 0

2ud 6

ekd

ek2

e

 2 (k )  2

2

Z 2 0

;

e

e 2 k 2 d

!h i p + ek  + 2! (  + ) d:

68

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

The last term in the right hand side of this inequality an be estimated as follows: !h Z 2 i 2 (e p  k ) e  e k2 2 + ek  + 2! ( + ) d 0

Z 2



 2

i h eku  + p2! (  + ) d + Z 0 2 h p i e e k ( + 2e  e k )  + 2! (  + ) d 2 Z0 2 u p h p i p 2e  ek  + 2! (  + ) d > 2 2 0 Z 2 Z h p i2 2 1 1 2 e 2 e k 2  + 2! (  + ) d u d 2 > 2  0 0 Z Z 2 2 1 1 e 2 ek2 d: > 2 u2 d 82  = 2



e

0

0

Finally, we arrive at Z 2 Z 2 Z 2 1 1 d 2 ud > d 0 2 0  u d + 2 0 ud 14

p Z 1 Z 2

2

Z 2 0

e

2

ek2 d + 4

Z 2 0

Lemma 3.14

2!

Proof:

dL d

0

0

()e ek(;  )dd 6 L(0) + ! 

dL dt dt d

=

p Z 2 2!

0

()e ek(;  )d 2!e 

ekd : (3.19)

Z 2

Integrate the equation =



e

2

0

j ()jd:

Z 2 0

 ( )d:



To dispose the last term on the right hand side of (3.19), we need the following lemma: Lemma 3.15

We have e

© 2001 by Chapman & Hall/CRC



ekmax( ) ! 0

as



!1 :

3.4.

69

The ontra ting ase of the ACEF

Proof:

From dL d

p

=

2!

2!e

Z

2

0 2

Z

()e  ke(;  )d 2

0

 ( )d ;

we have Z 1Z 0

2

0

e e k (;  )dd

6 C1 ;

(3.20)

for some onstant C1 . It follows from (3.13) and (3.20) that Z1 0

Fmax ( )e 2 d

6 C2

for some onstant C2 . As Fmax ( ) tends to in nity as  ! 1, we

an use Lemma 3.13 to show that Fmax is non-de reasing for large  . As a result, Fmax ( )e

2

6 2

Z1 

Fmax (s)e 2s ds

! 0 ; as



!1:

By this lemma, we nally obtain d d

Z

2 0

ud

© 2001 by Chapman & Hall/CRC

Z 2 Z 2 2 u 1 > 2  d + 2 0 ud 0 Z 2  Z 2 2 > Z1 ud ud + 2 0 0 2 d



70

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

2 Z

for all large  . As before, it means that ud 6 0 eventually, and 0

dE d

6 2 6 3

Z 2 0

Z 2 0

e

p  e kd + 2!

Z 2 0




e  ek  + d

e  ekd :

By (3.18),





E e(
;  )

E e(
; t) 6 3C1 :


We have shown that E e (
;  ) is uniformly bounded in [0; 1). As before, the bound on the entropy yields upper bounds for the diameter and length of the normalized ow, and also a positive lower bound for its inradius. Moreover, the normalized urvature and its gradient are also uniformly bounded. It remains to obtain a positive lower bound for the normalized

urvature. We shall do this by representing the ow as polar graphs. To this end, we rst observe that the urvature upper bound ontrols the speed of the ow. Therefore, there exists  > 0 su h that, for
all 0 , we an nd a disk D (x0 ; y0 ) en losed by e(
;  ) for all  2 [0 ; 0 + ℄. Using (x0 ; y0 ) as the origin, we represent e(
;  ) as

e(
;  ) = r(;  )( os ; sin ) ;  2 [0; 2) :

The tangent and normal of e are given, respe tively, by


t = r os  r sin  ; r sin  + r os  =D ; and

n=




r sin  + r os  ; r os  + r sin  =D ;

© 2001 by Chapman & Hall/CRC

(3.21)

3.4.

71

The ontra ting ase of the ACEF

where D = r2 + r2


1=2

. By dierentiating (3.21), we have


 e  r = Dt +

os  ; sin  :   

Therefore, r satis es

r =r 

Noti e that we also have

 D e p  k + 2!e  : r

(3.22)




sin  = r sin  + r os  =D ; and e k=

1

h

D3

i

rr + 2r2 + r2 :

By dierentiating (3.22), we obtain the equation for ek = ek(;  ):  ek    1  ek   3 ek + D =  D  D 

p

p




r ek + 2!e   rD

 e  p

i k e2 h + k  +  ek + 2!e    + ek :  2!eD   

During the time interval [0 ; 0 + ℄, r is bounded from above and below by positive onstants. Hen e, this equation is a uniformly paraboli equation with bounded oeÆ ients for ek. By the KrylovSafonov's Harna k inequality, or Moser's Harna k inequality [88℄ (noti e that it an be written in divergen e form), we on lude that e k > Æ > 0 for some Æ in [0 + =2; 0 + ℄. So ek has a uniform positive lower bound in [1; 1). With two-sided bounds on ek, we an easily nish the proof of the sub onvergen e of the normalized ow as follows. First of all, we may assume

p

ek + 2!e   > k0 > 0 ; for all  > 0 :

© 2001 by Chapman & Hall/CRC

(3.23)

72

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

Hen eforth, the ow is ontra ting and the support fun tion eh is positive if we x the shrinking point in the origin. We laim that, in fa t, eh > k0 =2 for all  . For, if eh < k0 =2 at some (0 ; 0 ), by (3.15) and (3.23), eh(0 ; 0 + 1) < 0, whi h is impossible. Now, onsider the fun tional

I ( ) =

Z 2 " e  h 2

# 2  e e d ; h + 2 log h + 2e



0

where  is to be hosen later. We have dI d

=

2 2

6

1 ek ee

e h

kh Z 2 0

Z 2 0

2

6

Z

0


2 e h +

kh

e 1

p

ek + 2!e

 

e h




 d

1 ek ee

Z 2

Z 2

e




Z 2 2! e 2 2 2 d e ke h 0

 d



e ke h

2
2 2! e h + 4 2 e  2 max

ek

k0





e 

6 0;

if we take  = 2!=k02 2 2max. Hen e, dI d

6

Z 2 0

1 ek ee kh


2 e h

60

(3.24)

along the normalized ow. From paraboli regularity theory, there is a uniform Holder bound on ek. From the boundedness of I and (3.24), we dedu e dI

e(
;  ) 0 = lim !1 d




© 2001 by Chapman & Hall/CRC

;

3.5.

73

The stationary ase of the ACEF

and

Z 2

lim  !1

0




ek eh 2 (
;  )d = 0 : 



We on lude that there exists a subsequen e e(
; j ) , j ! 1,

onverging smoothly to a self-similar solution of (3.10). The sub onvergen e of the normalized ow of (3.9) in the ontra ting ase holds without the positivity of . Besides, in the next hapter, we shall show that embedded self-similar solutions of (3.10) are unique up to homothety when  satis es ( + ) = (). Consequently, the normalized ow onverges smoothly to a self-similar solution in this ase. Remark 3.14

3.5

The stationary ase of the ACEF

In this se tion, we let (
; t) be a solution of (3.9) for  2 [ ;  ℄. We shall rst show that its urvature is uniformly bounded from below and above by positive onstants. A

ording to the de nition of  and  , we know that the length of (
; t), L(t), has a positive lower bound. We laim that it admits a uniform upper bound too. In fa t, let's ompute d  L2  dt A  = AL2 2A

Z 2 0

kd 2A


6 AL2 A L2  L2  6 L ; A A

© 2001 by Chapman & Hall/CRC

Z 2 0

 d + L

Z 2 0

Z



d + L ds

74

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

for some positive  and . Hen e,

n  L2 o L2 (t) 6 max ; (0) : A A

If L be omes very large, this estimate implies that we an nd a disk inside the urve. However, this is impossible sin e  >  . So L must be uniformly bounded from above. Noti e that the isoperimetri estimate also implies a positive lower bound for the inradius of (
; t) for all t. Lemma 3.15 The urvature is uniformly bounded between two positive onstants. Moreover, its derivatives are also uniformly bounded for all

t.

Let h be the support fun tion of a losed, onvex urve . We de ne its \support enter" (Chou-Wang [32℄) to be Proof:

=

Z 2

It is easy to see that

0

Rr

h()xd ; x = ( os ; sin ) :

in

> h()
x > rin ;

where rin is the inradius of and  is a positive absolute onstant. Let (t) be the support enter of (
; t). Consider the auxiliary fun tion h=t(; t) ; w(; t) = h(; t) (t)
x r0 =2 where rin > r0 > 0. Suppose the maximum of w over [0; 2℄  [0; T ℄ is attained at some point (0 ; t0 ), t0 > 0. At this point, we have


 2 h h h h + t t 







 2 h h  h + t2 t t

d 
x ; dt

0 = bh

Æ 2 w = (bh Æ)

0 6 bh

Æ 2 wt = (bh Æ)

© 2001 by Chapman & Hall/CRC

sin ; os 


i

;

3.5.

75

The stationary ase of the ACEF

and 0>

b h


2

=

Æ w

and bh = h


x

b h

Æ

 2h


3h

h + 2 t t

+
x  2



;

and Æ = r0 =2. Using the equation ht




=

k + 

;

we have  h 2

t

>

h d 2h h
x+ 2 b t dt t

>

 3h h d
x + k 2 t dt t 2

> k2

 2b h

+ bh  2

Æ

Æ

 h

t




+ h t



b h

Æ




d + h
x : t dt

In other words, w2 +

kw > k2 Æ w + d =dt
x ( b b b h

Æ

h

Æ

h

Æ

w) :

Using r0 6 bh 6 C0 , C0 some onstant, and d

dt

6 C1(1 + k) ;

we on lude that w, and, hen e, k, are uniformly bounded in [0; 1). This upper bound on the urvature yields an upper bound on the speed of the support enter. Using (t) as the origin, there exists
positive 4 and d0 independent of t su h that dist (t); (
; t ) > d0 for all t in [t; t + 4℄. Thus, we may represent (
; t ) as polar graphs and argue as in the previous se tion that k is uniformly bounded from below by a positive number, and also that there are uniform bounds on the derivatives of k. The proof of Lemma 3.15 is ompleted.  0

0

© 2001 by Chapman & Hall/CRC

0

76

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

The bounds we have obtained so far are not suÆ ient for onvergen e. In fa t, (
; t) may move to in nity in onstant speed. We shall show that onvergen e holds if and only if there is some fun tion  on S 1 satisfying 

 d2 



+ = : d 2

(3.25)

We modify the ow as follows. Let D=

n

( 1 ; 2 ) : = j j( os ; sin ) ; 0 6 j j
0, there exists   su h that

Proof:

max k (;   ) 6 ":  For, given " > 0, we divide [0; 2℄ into approximately [1="℄-many subintervals Ij s of length equal to ". For  > 0, we have Z  +1 ()ke(; s)ds = ( + eh)(;  + 1)e +1 + ( + eh)(;  )e 

and Z  +1 Z

()ek(; s)dds Z Z   +1 e ( + h)(;  + 1)d + e ( + eh)(;  )d; = e Ij



where

Z Ij

Ij

Ij

is the average over Ij . Summing over Ij s,

1 Z 2 Z  +1 0X Z  ()ek(; s)dA ds ()ek(; s)d + 0 I  j 3 2 Z Z 2 X ( + eh)(;  + 1)d + ( + eh)(;  + 1)d5 = e +1 4 I 0 j 3 2 Z Z 2 X ( + eh)(;  )d + ( + eh)(;  )d5 : +e 4 j

j

j

Ij

0

By the mean-value theorem, there exists   2 (;  + 1) su h that Z 2 XZ  ()k(;   )d ()k(;  )d + I 0 j    Z 2 Z 2  6 2e 1 + 1" j + ehj(;  )d + j + ehj(;  + 1)d 0 0  e e + max fj + hj(;  ) + j + hj(;  + 1)g :  j

© 2001 by Chapman & Hall/CRC

84

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

Sin e by Lemma 3.19,  + eh tends to zero uniformly, for suÆ iently large  we have X

Z

Ij

j

()k(;   )d +

Z 2

0

()k(;   )d < ":

Suppose that max k(;   ) is attained at  = 0 , 0 in some Ij . We have



6

k





Z

max k(;   )

Ij

k (;   )d

(
;  )



L1

"

6 C" by Lemma 3.13. Therefore, max k (;   ) 6 (1 + C )": 

Now, we an apply Lemma 3.17 to , using  =   as the initial time, to show that ek is uniformly bounded.  To show higher order onvergen e, we look at the equation satis ed by w =  (  + ) e k: By a dire t omputation where B C

w 

=e

 e2

k

2

 w2 + e



B

w 

+e



Cw + w(1

w) ;

(3.31)

= 2  + ( (  ++ ) )  ;   + )  + 2(  + )2  + (  + )  = ek2 (  ( + ) ( + )2 ( + ) 



e k2



 + ) + (2 ( 2  + )

w2

© 2001 by Chapman & Hall/CRC



#

:



3.6.

85

The expanding ase of the ACEF

Therefore, by Lemma 3.20 we have dwmax d dwmin d

6 Ae  + wmax(1 wmax); and >

Ae



+ wmin(1 wmin)

for some onstant A. It follows that w tends to 1 uniformly as  approa hes 1. By dierentiating (3.31), we see that u = w= satis es the equation   2u C u u  0 0 0 = e A 2 + B + C u + e  w + (1 2w)u     where A0 , B 0 , and C 0 are uniformly bounded in [0; 2℄  [0; 1). Sin e w tends to 1 uniformly, it is easy to see that, for any  2 (0; 1), w (
;  ) = O(e 

 )

uniformly as  ! 1:

Higher order estimates an be obtained in the same way. The proof of Proposition 3.16 is ompleted. Finally, we want to show that (3.28) is ne essary for C 2 - onvergen e of eh. More pre isely, we have, Proposition 3.21

If

eh(
; t) onverges to some eh in C 2-norm, then

(3.28) holds.

From (3.30),   dekmin e  2 e e > kmin 1 + kmin min (  + )+ e kmin min( + ) ;   d

Proof:

we an see that ekmin has a positive lower bound. When eh(
; t) tends to eh in C 2 -norm, the urve determined by eh must have positive urvature. However, by Lemma 3.19, eh is equal to  . Hen e, (3.28) holds. 

© 2001 by Chapman & Hall/CRC

86

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

Notes Gage-Hamilton Theorem.

Theorems 3.4 and 3.5 were proved in [58℄. The notion of entropy was impli itly introdu ed and its monotoni ity property was established in this paper. To show the

onvergen e to a ir le, they used an inequality of Gage [54℄ (see x4.4) whi h asserts that the isoperimetri ratio de reases along the

ow for onvex urves. In another proof [72℄, Hamilton appeals to the hara terization of the self-similar solution due to Abresh and Langer. Finally, Andrews [10℄ dedu ed the onvergen e from a hara terization of equality in the entropy inequality whi h follows from the Minkowski inequality. Our proof here is elementary and is based on the monotoni ity of the fun tional F (Firey [51℄) and Proposition 2.3. The models.

The CSF was posed as a phenomenologi al model for the dynami s of grain boundaries in Mullins [?℄ in the fties. Sin e then, it and the more general ACEF have arisen in many different areas, in luding phase transition, rystal growth, ame propagation, hemi al rea tion, and mathemati al biology. We brie y dis uss three of these areas. First, in the sharp interfa e approa h to the theory of phase transition, dierent bulk phases are separated by a sharp interfa e whi h is assumed to be a plane urve. The temperatures in both bulks satisfy the heat equation and the interfa e is a free boundary. However, in the ase of a perfe t ondu tor, the temperatures are

onstant in both phases and it is possible to redu e the motion of the interfa e to a single equation. This was done in Gurtin [71℄. In fa t, he rst derived a balan e law relating the apillary for e of the interfa e and the intera tive for e. Next, he used a version of the se ond law of thermodynami s to obtain onstitutive restri tions on

© 2001 by Chapman & Hall/CRC

3.6.

87

The expanding ase of the ACEF

the form of the equation. The resulting law of motion of the interfa e is b( )

 t




= (f + f )k + F n ;

where F is a onstant (the dieren e in energy between the phases), b > 0 a material fun tion that hara terizes the kineti s, and f the interfa ial energy. The dependen e of b and f on the normal angle  re e ts anisotropy. In ase f + f > 0, it was proved in AngenentGurtin [17℄ that (a) if F > 0, then ! < 1 and the en losed area A(t) ! 0 as t " ! ; and (b) if F < 0, then (i) if the initial length is suÆ iently small, then ! < 1 and A(t) ! 0 and (ii) if the initial area is suÆ ient large, then ! = 1, A(t) ! 1 and the isoperimetri ratio remains bounded. They also onje tured that in ase (b)(ii), the ow is asymptoti to a Wul region of 1=b(). This onje ture was subsequently on rmed by Soner [100℄. Our Theorem 3.12 (Chou-Zhu [35℄) gives a pre ise and omplete des ription of the

ow for onvex urves. We point out that the ow may develop selfinterse tions when it is non- onvex. In this ase, one has to use the level-set approa h ([100℄). When  0, the ow was also studied in Gage [57℄ and Gage-Li [59℄ in the ontext of Minkowski geometry. In parti ular, Theorem 3.12(i) was proved in [59℄ for this ase. One should keep in mind that the model also makes sense when f + f 6 0 or when it is not C 2 . As a spe ial important ase, we

omment on the rystalline energy whi h, by de nition, has a polygonal Frank diagram (the Frank diagram is the polar graph of 1=f ()). Its derivatives have jumps at the verti es. The motion laws for the

rystalline energy were derived by Angenent-Gurtin [17℄ and Taylor [106℄ independently. This is a system of ODEs on the sides of the polygon under evolution. Crystalline versions of the Gage-Hamilton theorem and the Grayson onvexity theorem an be found in Stan u

© 2001 by Chapman & Hall/CRC

88

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

[103℄, [104℄ and Giga-Giga [62℄ respe tively. For a level-set approa h, see also Giga-Giga [61℄. Se ond, in the eld equation approa h to phase transition, one

onsiders rea tion-diusion equations su h as the Allen-Cahn equation, ut = 4u

W 0 (u) ; (x; t) 2 Rn

 (0; 1) ;

(3.32)

where W is a double-well potential having exa tly two stri t minima, say, at u = 1 and 1. The states u = 1 represent stable phases. It is well-known that (3.32) admits a unique travelling wave solution (x1 t; x2 ;


; xn ), satisfying (1) = 1 and 0 > 0 where the wave speed is given by Z 1
1
2

= W ( 1) W (1) : 1 0

When = 0, the CSF arises as the singular limit of a general solution of (3.32) as t ! 1. In fa t, let u"(x; t) = u(x="; t="2 ). Then u" satis es u"t = 4u"

"

2

W 0 (u" ) :

(3.33)

Let's fo us on n = 2 and let 0 = fu0 (x) = 0g and D0 = fu0 (x) > 0g for a given initial u0 . In DeMottoni-S hatzman [40℄ and Chen [27℄, the following result is proved: Theorem. Let u" be the solution of (3.33) satisfying u"(x; 0) = u0 . Let (
; t) be the CSF starting at 0 and Dt the region it en loses. Then as " ! 0, u"

© 2001 by Chapman & Hall/CRC

[

8 0

S

(Dt  ftg)

(R2 n Dt  ftg)

3.6.

The expanding ase of the ACEF

ompa tly in

89

R.

So the interfa e moves approximately under the CSF when time is long enough. In identally, we mention that, when the wave speed is non zero, one should take the s aling u" (x; t) = u(x="; t=") and the interfa e moves approximately under the eikonal equation t = n. Higher dimensional results of this kind an be found in Souganidis [101℄ and the referen es therein. Finally, the (isotropi ) urvature-eikonal equation was derived in the study of wave propagation in weakly ex itable media (KeenerSneyd [84℄ and Meron [89℄). (The name urvature-eikonal equation is taken from [84℄.) This equation is relevant in a number of biologi al and hemi al ontexts, su h as the wave front propagation in the ex itable Belousov-Zhabotinsky reagent, the al ium waves in Xenopus oo ytes, and the studies in myo ardial tissue. Here the rea tion diusion system is of the form

(

4 4

"ut = "2 u + f (u; v ) vt = "D v + g (u; v ) ;

where u is the a tivator (propagator) and v is the inhibitor ( ontroller) in the medium. A spe i example of the rea tion term is the FitzHugh-Nagumo dynami s. When " is very small, one an formally show that the system an be des ribed by

(

t = ("k + (v ))n vt = "D v + g (u (v ); v ) ;

4

where u (v) are the two stable roots of f (u; v) = 0 (Keener [82℄). In general, the rst equation in this system des ribes a free boundary while the se ond equation should be satis ed by v in the two separating regions. We obtain the urvature-eikonal ow when is independent of v. In general, one expe ts this oupled system to be useful in explaining ertain ommonly observed stable patterns, su h

© 2001 by Chapman & Hall/CRC

90

CH. 3.

The Curvature-Eikonal Flow for Convex Curves

as the spirals, in dynami s in ex itable media. The CSF and the urvature-eikonal ow are also relevant in other physi al ontexts, see, e.g., De kelni k-Elliott-Ri hardson [42℄ for the motion of a super ondu ting vortex, and Frankel-Sivahinsky [52℄ for the propagation of ame front.

The level-set approa h in urvature ows. This approa h provides a unique, globally de ned generalized solution for many geometri ows, in luding the ACEF. Although the results are not only valid for urves but also for many geometri ows, in luding the anisotropi nisotropi mean urvature ow in higher dimensions, for the purpose of illustration we state them for urves only. To start, let's onsider the CSF. Assume that there is a fun tion u(x; t) whose level set (
; t) = fu(x; t) = g satis es the CSF for ea h xed and that fu(x; t) > g is the bounded set en losed by (
; t). Then we have n = Ou=jOuj and t
n = ut =jOuj on (
; t). Therefore, u satis es the equation

Ou
: Ou

ut = jOuj div

j

(3.34)

j

Conversely, one an verify that, for any solution of (3.34), the set

(
; t) = fu(x; t) = g satis es the CSF. Consequently, one may use this onne tion to de ne a generalized solution of the CSF whenever a solution of (3.34) is given. The equation (3.34) is weakly paraboli and is not de ned at jOuj = 0. It is ru ial that, nevertheless, a notion of generalized solution, namely the vis osity solution, is available for this equation. It turns out this onsideration works for other

ows. To state a sample result, let's onsider (1.2) where F depends on  and q only. The orresponding equation is ut = jOujF

© 2001 by Chapman & Hall/CRC

 j

Ou Ou ; j

div

Ou Ou

j

j



:

(3.35)

3.6.

The expanding ase of the ACEF

91

Denote by C (R2 ) the family of ontinuous fun tions u su h that u  is of ompa t support. We have (Chen-Gigo-Goto [28℄): Theorem. Assume that F = F (; q) is uniformly paraboli in (1.2). Let D0 be an open set and 0  D0 be a bounded set. Then for any fun tion u0 in C (R2 );  < 0; whi h satis es fu0 > 0g = D0 and fu0 = 0g = 0 , there exists a unique vis osity solution of (3.35) satisfying u(
; 0) = u0 . Moreover, the sets (
; t) = fu(x; t) = 0g and Dt = fu(x; t) > 0g depend only on 0 and D0 but not on  or u0 . Consequently, the ow (1.2) admits a unique generalized solution for all t. Noti e that in some ases su h as the CSF, (
; t) be omes empty for t > !. A mathemati al theory on the level-set approa h was introdu ed in [28℄ and Evans-Spru k [50℄ independently. In [50℄ and its sequels, the mean urvature ow is dis ussed in some depth, while in [28℄ other geometri urvature ows are also treated. For further work, we refer to Giga-Goto-Ishii-Sato [63℄, [100℄, and the survey [101℄. For a geometri measure-theoreti approa h to the mean urvature ow, one may onsult Brakke [22℄ and Ilmanen [81℄. See also Ambrosio-Soner [7℄ for another approa h by De Giorgi.

© 2001 by Chapman & Hall/CRC

Chapter 4

The Convex Generalized Curve Shortening Flow In this hapter, we study the Cau hy problem for the anisotropi generalized urve shortening ow (AGCSF),  = ()k n ;  > 0 ; (4.1) t where  is a smooth positive, 2-periodi fun tion of the normal angle and 0 is a given uniformly onvex, embedded losed urve. A

ording to the results in x3.2, this ow preserves onvexity and shrinks to a point in nite time. To examine its ultimate shape, we normalize the ow using the shrinking point as the origin so that its en losed area is always equal to . In terms of the support fun tion and urvature,  A(t)  1=2 e h; and h =  e k

=

 A(t) 1=2



k;

we have the equations  Z  e e e  k  1 d e h; h = k + S1 and Z h i
e ek k = e k 2 e k   + e k 1 S

93 © 2001 by Chapman & Hall/CRC

(4.2) 1




k; d e

(4.3)

94

CH. 4.

The Convex Generalized Curve Shortening Flow

where d =dt = (A(t)=) (1+)=2 . It will be shown, (see Lemma 4.6), that  = O(log(! t)). Hen e, (4.2) and (4.3) are de ned in [0 ; 1) for some 0 . Formally, in ase eh tends to zero as  ! 1, the limit satis es the equation k = h ;

(4.4)

where  is a positive onstant. In other words, it is a ontra ting self-similar solution for (4.1). We shall establish the sub onvergen e of the normalized ow to self-similar solutions for  2 (1=3; 1) in Se tion 2. Some useful inequalities from the theory of onvex bodies are listed in Se tion 1. In Se tion 3, we study the aÆne urve shortening problem, that is,  = 1=3 and   1 in (4.1). It has an invariant formulation in the ontext of aÆne geometry. We shall show that onvergen e to ellipses holds for the normalized ow. In the last se tion, we present a uniquen e result on self-similar solutions for (4.1) where  > 1 and  is symmetri . 4.1

Results from the Brunn-Minkowski Theory

In this se tion, we olle t some basi results from the Brunn-Minkowski Theory of onvex bodies. Although the natural setting is onvex bodies, for our purpose it is suÆ ient to state them for onvex bodies with a smooth boundary. As a result, they will appear in slightly restri tive form. Let h and ' be two smooth fun tions de ned on S 1 . We shall use the following notations: A0

© 2001 by Chapman & Hall/CRC

=

Z S1

hA[h℄d ;

4.1.

95

Results from the Brunn-Minkowski Theory

A1 A01

= =

Z S1

Z S1

'A['℄d ; 'A[h℄d

=

and Z S1

hA['℄dd ;

where A[f ℄ = f + f for any fun tion f . Theorem 4.1 (Minkowski inequality). We have

A201

> A0A1

under either one of the following onditions: (i) A[h℄

0,

> 0 and A['℄ >

or (ii) A[h℄ > 0. Moreover, equality in this inequality holds if and

only if there exist a positive  and a point (x0 ; y0 ) su h that

'( ) = h( ) + for all 

h(x0 ; y0); ( os ; sin )i ;

2 S1.

Theorem 4.1 under assumption (i) is the standard form of the Minkowski inequality and usually is dedu ed from the Brunn-Minkowski inequality. Under (ii), we may apply the inequality to the fun tions h and ' + h, where  has been hosen so large that A[' + h℄ > 0. The following two results are about stability. They provide effe tive lower bounds of the dieren e

4 = A201

A0 A1 :

Let K0 and K1 be two onvex sets with support fun tions h and ', respe tively. The ir umradius and inradius of K0 with respe t to K1 are given by n R(K0 ; K1 ) = inf R > 0 : K0

© 2001 by Chapman & Hall/CRC

R

K1




(x; y) for some (x; y)

o

96

CH. 4.

The Convex Generalized Curve Shortening Flow

and n




r (K0 ; K1 ) = sup r > 0 : r K1

(x; y)

o

 K0 for some (x; y)

;

respe tively. When K1 is the unit disk, the ir umradius and inradius of K0 are the usual ir umradius and inradius of a onvex body (or its boundary). Our rst stability estimate is an anisotropi version of the Bonnesen inequality.

Theorem 4.2

Let

R = R(K0 ; K1 )

 2 [r; R℄.

r = r (K0 ; K1 ).

We have

2A01 + 2 A1 6 0

A0 for all

and

Consequently,

2

4 > A41 (R

r )2 :

0 and K

1 , Next, let's denote by bh and 'b the support fun tions K whi h are obtained from K0 and K1 by translating their enters of mass to the origin and res aling their perimeters to 1. We have:

Theorem 4.3

There exists a onstant

ameter and inradius of

K0

4>

C

depending only on the di-

su h that

CA201

Z S1

(bh

'b)2 d :

Finally, we state two isoperimetri inequalities somehow related to the aÆne and SL(2; R ){geometries:

Theorem 4.4 (The aÆne isoperimetri inequality)

.

losed embedded onvex urve

Z S1

k

2=3

,

d


3

we have

6 82 A ;

and the equality holds if and only if

© 2001 by Chapman & Hall/CRC

is an ellipse.

For any

4.2.

The AGCSF for



in

(1=3; 1)

97

As will be explained in Se tion 3, the integral on the left-hand side of this inequality is the aÆne perimeter L of . Thus, this inequality asserts that only the ellipses maximize the aÆne isoperimetri ratio L3 =A. Theorem 4.5 (Blas hke-Santal o inequality). embedded onvex urve

,

Z

there exists a point

1 d 6 22

2 S1 h

where

h

A

is the support fun tion of

For any losed

(x0 ; y0 ) su h that

;

with respe t to

(x0 ; y0 ).

more, equality in this inequality holds if and only if

4.2

The AGCSF for



in

Further-

is an ellipse.

(1=3; 1)

In this se tion, we generalize the Gage-Hamilton Theorem to (4.1) for  2 (1=3; 1). As we have seen in the previous hapter, the monotoni ity of the entropy is essential in ontrolling the urvature of the normalized ow. Fortunately, a de nition of entropy is available and monotoni ity along the ow is still valid. We de ne the entropy for the normalized ow (4.2) to be


E e (
;  ) =

Z S

()ek 1

1

d



1

 1

;  6= 1 :

When  = 1, the entropy was de ned in (3.17). Lemma 4.6

For all

 > 0,


d E

e(
;  ) 6 0 ; dt and the equality holds if and only if of (4.1) whi h ontra ts to some

© 2001 by Chapman & Hall/CRC

(
;  )

(x0 ; y0 ).

is a self-similar solution

98

CH. 4.

Proof:

=

The Convex Generalized Curve Shortening Flow

By (4.3),


d E

e(
;  ) d Z 1  ()ek 1 d 2 2

hZ 1

1  Z ek A[eh℄d2 i : 2A

ek A[ek ℄d

Noti ing that

Z 1 e e A= 2 hA[h℄d is equal to , it follows immediately from the Minkowski inequality that dE =dt( e(
,  )) 6 0. Moreover, in ase equality holds at some  , then e(
;  ) satis es (4.4), where h is de ned with respe t to some (x0 ; y0 ).  Lemma 4.7 For any diameter of



in

(1=3; 1),

the isoperimetri ratio and the

e(
;  ) are uniformly bounded by a onstant depending on

its entropy.

It suÆ es to show that the width w() of e(
;  ) along any dire tion  has a uniform, positive lower bound. Consider  > 2=3 rst. By the Holder inequality,

Proof:

Z

ke 1 d

6

Z

e k 1 os(




1 

) d

Z

os(



1




) 1  d

:

Therefore, w() > C0 E ( e(
;  ))

1

:

Next, for  2 (1=3; 2=3), by the Holder inequality again,  1

2



E ( e(
;  )) > E 23 ( e(
;  )) 1 3 E 31 ( e(
;  )) 31  : By the aÆne isoperimetri inequality and the de nition of E 31 , we have

E 13 ( e(
;  )) > (2min)

© 2001 by Chapman & Hall/CRC

3=2

:

4.2.

The AGCSF for



in

(1=3; 1)

99

Therefore,

E ( e(
; 0)) > (2min) >

C0 (2 min )

E 3 (
;  )

3=2 w

1 3




3=2 2 e

1

() 31



1

 

:

 Lemma 4.8

For



2 (1=3; 1), the urvature of e(
;  ) is uniformly

pin hed between two positive onstants.

We rst derive an upper estimate. Consider the auxiliary fun tion  = ( ht )=(h (t)
x Æ) as in the proof of Lemma 3.15. Re all that (t) is the support enter of h and Æ = r0 =2. By following the same proof, we arrive at the estimate Proof:

k

6 C (h

(t)
x

Æ) :

However, sin e now the isoperimetri ratio of (
; t) is uniformly bounded by Lemma 4.7, there exists a onstant C 0 su h that h

(t)
x  C 0 rin (t). Consequently, e k  is bounded from above. Next, by the gradient estimate in Lemma 3.7, whi h holds for (4.3) after repla ing ek by ek and is s aling invariant, we know that  1 1   (e (e k ) = k ) ; se





where se is the ar -length of e. Sin e the length of e(
;  ) is uniformly bounded and  < 1, by integrating this inequality we obtain a positive lower bound for ek.  Theorem 4.9

For



2 (1=3; 1), any sequen e f e(
; j )g, j ! 1,

ontains a subsequen e whi h onverges smoothly to a ontra ting self-similar solution of (4.1) .

© 2001 by Chapman & Hall/CRC

100

CH. 4.

The Convex Generalized Curve Shortening Flow

We have obtained two-sided bounds on the normalized urvature in the pre eding lemmas. By representing the equation for ek as polar graphs we an obtain all bounds on the derivatives of ek (see x3.4). Therefore, by Lemma 4.6, Proof:

dE lim ( e(
;  )) = 0  !1 d

and any sequen e f e(
;  )g ontains a subsequen e onverging smoothly to a onvex urve 1, whi h, a

ording to the Minkowski inequality, satis es  k1 = h1 + h(x0 ; y0 ); ( os ; sin )i

for some positive  and (x0 ; y0 ). If (x0 ; y0 ) is not the origin, from A(t)1=2 e =  1=2 we see that, for all t suÆ iently lose to ! , (
; t) no longer ontain the origin. But this is impossible. Hen e, (x0 ; y0 ) must be the origin. The proof of Theorem 4.9 is ompleted.  When  > 1, the isoperimetri ratio and the urvature of e(
;  ) are still bounded from above. This follows from

ombining the entropy inequality and the s aling invariant gradient estimate in Lemma 3.7 (repla e k by k ). However, a positive lower bound for the urvature may not be available. Instead, one an derive a gradient estimate for the urvature dire tly using (4.3). In [10℄, the following theorem is proved.

Remark 4.10

Theorem

For

 > 1,

any

f e(
; j )g; j ! 1, ontains a subsequen e

whi h onverges to a self-similar solution of (4.1) in where

= ( 1) 1

k 2 (0; 1℄.

C k+2; -norm

Moreover, the onvergen e is smooth

away from points where the urvature of the self-similar solution is zero.

An example in the same paper shows that the self-similar solution may not be stri tly onvex and the regularity on erning the sub onvergen e is optimal.

© 2001 by Chapman & Hall/CRC

4.2.

The AGCSF for



in

(1=3; 1)

101

Where  < 1=3, Theorem 4.9 holds under the additional assumption that the isoperimetri ratio of e(
;  ) is uniformly bounded. Again, this assumption annot be ful lled most times. To see why, we look at the equation for the self-similar solution (4.4) whi h is now written in the form of an ellipti equation , a( ) ; (4.5) h00 + h = Remark 4.11

hp

where a() = ( 1 )1= and p = 1=. For simpli ity, we shall onsider the isotropi ase (a()  1) only. Self-similar solutions are

riti al points of the fun tional Z Z 1 2 2 F (h) = 2 1 (h h )d 1 h1 pd : S S Along the normalized ow, we have d e F (h(
;  )) dt

= (1 = (1

By the Holder inequality, Z

and

e k

Z

h





1 d 6 1

d 6

p) p)

Z

Z

Z

e h

Z

pe h d

Z

1 d eh p+1 d

e k

e k h

p

e k h

p

d

d




+1


1

+1

Z

Z

Z

e h

k d ) :

p e

1 h
+1 d k h
+1 : d k

Therefore, for  6 1, d e F (h(
;  )) > 0 ; dt

and the equality holds if and only if eh is a self-similar solution. When  > 1=3, the Blas hke-Santalo's inequality asserts that

© 2001 by Chapman & Hall/CRC

102

CH. 4.

The Convex Generalized Curve Shortening Flow

the fun tional I has a universal upper bound. However, F be omes unbounded if  < 1=3. To see this, let's onsider the boundary of a re tangle ( entered at the origin) whose dimension is 2` by =2`. Its support fun tion is given by

r h=

`2 +

2 `2

8 > < os(0 > :

os(

; 0 6  6 0

) 0 )

; 00

6  6 =2 :

One an verify dire tly that, for p > 3,

Z

S1

h1 p d

tends to 1 as ` ! 1. On the other hand, one an show that, in the isotropi ase, ir les are the only self-similar solutions (or at least show that sup I over all self-similar solutions is nite). Therefore, for any initial urve whose support fun tion h0 satis es F (h0 ) > supfF (h) : h is self-similarg, the normalized ow starting at h0 never sub onverges to a self-similar solution. In fa t, its isoperimetri ratio tends to in nity as  ! 1.

4.3 The aÆne urve shortening ow We begin with a very brief des ription of the aÆne geometry of plane

urves. Let be a plane urve. A parametrization of , , is alled the aÆne ar -length parameter if it satis es

2

1 [  ;  ℄  det 6 4

3 7 5

2

2 1





= 1

© 2001 by Chapman & Hall/CRC

4.3. The aÆne urve shortening ow

103

along the urve. For any uniformly onvex urve with a given parametrization p, its aÆne ar -length always exists and is given by Zp [ p ; pp ℄1=3 dp : = 0

Noti e that, when is parametrized by the Eu lidean ar -length s or the normal angle , the aÆne ar -length is given by Z  = k 1=3 ds

or =

Z S1

aÆne ar -length of

The is given by

L

( ) =

k

2=3

d :

aÆne perimeter when is losed)

(

Z

d :

It has the invarian e property, namely, L

( A ) = L( ) ;

for any A  A
where A 2 SL(2; R ) is any spe ial aÆne transformation in the plane. The and of are given by J =  and N =  , respe tively. They satisfy AJ = JA and AN = NA where JA and NA are, respe tively, the aÆne tangent and normal for A . By dierentiating [  ,  ℄ = 1, we see that  and  are linearly dependent and, hen e,  +   = 0 for some . The fun tion

aÆne tangent

aÆne normal

 = [  ;  ℄

aÆne urvature

is the of . It turns out that the aÆne urvature is an absolute invariant of the spe ial aÆne group SL(2; R ) in the

© 2001 by Chapman & Hall/CRC

104

CH. 4.

The Convex Generalized Curve Shortening Flow

sense that the aÆne urvature of A at A (
) is equal to the aÆne

urvature of at (
). Any fun tion with this property is alled an absolute invariant or a dierential invariant. Other absolute invariants of SL(2; R ) in lude the derivatives of the aÆne urvature with respe t to the aÆne ar -length, as one an he k dire tly. In fa t, any absolute invariant of SL(2; R) must be a fun tion of  and its derivatives. Just as the Eu lidean urvature determines the urve up to a Eu lidean motion, the aÆne urvature also determines the

urve up to a spe ial aÆne transformation.

Example 4.12 Curves of onstant aÆne urvature.

As one

an he k dire tly, parabolas have zero aÆne urvature, and ellipses and hyperbolas have positive and negative aÆne urvature, respe tively. Let () = (a os , b sin ) where  = (ab)1=3 . Then,  is the aÆne ar -length parameter and the aÆne urvature is given by  = (ab) 2=3 . Similarly, let C () = (a osh( ), b sinh( )) where  = (ab)1=3 . Then,  is the aÆne ar -length parameter and  = (ab) 2=3 . The area en losed by the losed urve remains un hanged under all spe ial aÆne transformations. Hen e, the quotient Z 2=3 3 L3( ) = ( S1 k d) A

A

remains un hanged under all general aÆne transformations, that is, the full aÆne group GL(2; R). The aÆne isoperimetri inequality asserts that only the ellipses maximize this quotient. Now, let's onsider the ow d =N ; dt

(4.6)

where N is the aÆne normal of (
; t). Sin e NA = AN , we see that, if (
; t) solves (4.5), so does A (
; t) for any A 2 SL(2; R). In

© 2001 by Chapman & Hall/CRC

4.3. The aÆne urve shortening ow

105

parti ular, all ellipses are ontra ting self-similar solutions for (4.6). Given (4.6), one may derive the orresponding evolution equations for the geometri quantities of the ow. Sin e the derivation is somehow parallel to the Eu lidean ase whi h was done in x1.3, we list the result only. One may onsult [98℄ for details. We have t

=

2  ; 3

Tt

=

1 T ; 3

Nt

=

1 N 3

(4.7)

1  T ; 3 s

and 4 1  + 2 : 3  3

(4.8)

At =

L and

(4.9)

Lt =

2 3

t =

We also have

Z

d :

(4.10)

Sometimes it is onvenient to express  and  as fun tions of the normal angle . In fa t, we an verify that d = v 2 d

and  = v 3 (v + v ) ;

where v  k1=3 . Let's express the aÆne normal in terms of the Eu lidean tangent

© 2001 by Chapman & Hall/CRC

106

CH. 4.

The Convex Generalized Curve Shortening Flow

and normal. We have N

= k

1=3

s

= k

1=3

(k

1=3

= k1=3 n + k

s )s

1=3

(k

1=3

)s t :

Hen e, by Proposition 1.1, the ow (4.6) is equivalent to (4.1) with  = 1=3 and   1. The normalized ow e(
;  ) of (4.1) ,  = 1=3 and   1, onverges smoothly to an ellipse entered at the origin.

Theorem 4.13

Lemma 4.14

We have d 3 L ( e(
;  )) > 0 ; d

and equality if and only if e(
;  ) is an ellipse.

This is nothing but the monotoni ity property of the entropy in Lemma 4.6 for  = 1=3.  Proof:

Let 1 2 (0 ; 1) be xed. We hoose a spe ial aÆne transformation A su h that 0

= A e(
; 1 )





is bounded between the ir les CR 1 (0; 0) , and CR (0; 0) where R is an absolute onstant greater than 1. (In fa t, one an take
R = 3.) Observe that the ir le CR 1 (0; 0) shrinks to a point
under the aÆne ow. Let T be the time it rea hes C1=(2R) (0; 0) . By the strong separation prin iple, we know that (
; t), the aÆne ow

starting at 0 , is bounded between C1=(2R) (0; 0) and CR (0; 0) for

© 2001 by Chapman & Hall/CRC

4.3. The aÆne urve shortening ow

107

all t in [0; T ℄. Letting H be the support fun tion of , we onsider the auxiliary fun tion tH (; t) = ; t 2 [0; T ℄ ; H (; t)  where  = 1=(4R). Arguing as in the proof of Proposition 3.3, we obtain the inequality K 5=3 t0 4 K 2=3 t0 6 + K 1=3 + K 2=3 ; 3(H ) 3 H  where K (the urvature of ) and H are evaluated at (0 ; t0 ), a maximum of . It follows that K (; t) 6

C ; t

8(; t) 2 S  (0; T ℄ : 1

In parti ular, K has a uniform upper bound on [T =2; T ℄. For any large  , we an nd 1 <  , 1 lose to  su h that the ow (
; t) starting at A (
; 1 ) as des ribed above satis es e (
;  ) = A 1 (
; T 0 )
for some T 0 in [T =2; T ℄. Sin e (
; T 0 ) is bounded between C1=(4R) (0; 0)
and CR (0; 0) , we an multiply it by a onstant so that its en losed area is . We have shown that, for ea h large  , there exists A 2 SL(2; R) su h that (
;  ) = A e

(
;  ) has uniformly bounded

urvature. Next, we laim that k, the urvature of , also admits a positive lower bound when  is suÆ iently large. First, we note that, by applying the maximum prin iple to (4.8), (
; t) > inf (
; 0) ;

i.e.,  e(
;  ) >

 A(t)  2

3



inf (
; 0) :

Using the aÆne invarian e of the aÆne urvature, the aÆne urvature of , satis es (
;  ) >

© 2001 by Chapman & Hall/CRC

C0 A(t)2=3 :

108

CH. 4.

The Convex Generalized Curve Shortening Flow

We rewrite this inequality as A[V

+ C0 A(t)2=3 H ℄ > 0 ;

where we have set V = k 1=3 . Hen e, H  V + C0 A2=3 (t)H de nes a

onvex urve . By the formula Z 1 d ; L( ) = S 1 K (;  ) we dedu e that for any  > 0,  K

6



6

L(

6



)

 2R :

Noti e that, by onvexity, the perimeter of is less than the perime
ter of CR (0; 0) . It follows from the de nition of H that  H



6  6 3 (2R)3 :

(4.11)

On the other hand, let (0 ;  ) be a point in the set We have, for all  satisfying os( 0 ) > 0, H ( )

6  os(

0 ) + d

p

1 os(

fH < g.

0 )2 :

This is be ause the right-hand side of this inequality is nothing but the support fun tion of the region lying inside a ylinder of width d and with axis along ( os 0 ; sin 0 ), trun ated by the line f os 0 x + sin 0 y = g. When d is larger than the maximal width of , this region ontains and, hen e, the inequality holds. It follows that n  : sin(

and so

© 2001 by Chapman & Hall/CRC

0 )

 H

o

6 

n



H

6 (1 + d)

6 (1 + d) > 2 :

o

;

(4.12)

4.3. The aÆne urve shortening ow

109

By putting (4.10) and (4.11) together, we see that the set fH 6 g is empty if  < 21=2 (1 + d) 3=2 (2R) 3=2 . From k

1=3

=V >

( )2=3 H + 2

C0 A t

3=2

R

3=2

;

we dedu e a positive lower bound for k when t is suÆ iently lose to ! . We remark that su h a bound an also be derived from the upper bound on K by using the Harna k's inequality of Chow [37℄. With two-sided bounds on K , we an use paraboli regularity theory to obtain bounds on the derivatives of K (in ) for all suÆ iently large 1 . We have proved the following lemma. For ea h  , there exists A 2 SL(2; R ) so that (
;  ) = A e(
;  ) satis es (a) its urvature is uniformly pin hed between two positive onstants and (b) the derivatives of its urvature are uniformly bounded.

Lemma 4.15

We note that three useful messages are en oded in this lemma. First, by aÆne invarian e, the aÆne urvature of e, as well as its derivatives with respe t to the aÆne ar -length, is uniformly bounded. Next, sin e by Lemma 4.14 we know that there exists a sequen e f (
; j )g, j ! 1, su h that (
; j ) tend to 1, e(
; j ) also tends to 1. Further, by applying the maximum prin iple to (4.8), we on lude that (; t) > inf (; tj ) ; t > tj ;  and so, (
; t) is positive for all t lose to !. lim !1e(
;  ) = 1. Consider the fully aÆne invariant quantity

Lemma 4.16 Proof:



L (t )

© 2001 by Chapman & Hall/CRC

Z

d :

110

CH. 4.

The Convex Generalized Curve Shortening Flow

Noti e that, by the Minkowski inequality, Z 1 L2=A ; d 6 2 Z 3 L d 6 21 LA 6 42 ;

and equality holds if and only if the urve is an ellipse. By (4.7), (4.8), (4.10), and the Holder inequality, Z Z Z
  d 2 2 L ( t) d = ( d ) + L 2 d dt 3 > 0:

Hen e in the equality Z 1 L2 = A L Z d L3
; A d 2 L 2A R both terms L d and L3=(2A) in rease to 42 as t " !. We on lude 1 Le2  Z ede ! 0 as  ! 1 : 2 By aÆne invarian e, we also have 1 L2  Z d = o(1) 2

as  ! 1. Now, let's apply Theorem 4.3 to the fun tions h = h and 1=3 '=k to get Z 1=3 lim j h k j2 d = 0 :  !1 In view of Lemma 4.15, we dedu e 1=3 lim hk =1: (4.13)  !1

© 2001 by Chapman & Hall/CRC

4.3. The aÆne urve shortening ow

111

Now, we onvert the information on to an estimate on e as follows. One an verify dire tly that hk

1=3
 

+ (h k

1=3

)=1:

By (4.13) and interpolation, lim !1(h k

1=3



)  = 0 :

Therefore, lim !1 e = lim !1  = 1. The proof of Lemma 4.16 is ompleted.  Now we an prove Theorem 4.13. By a dire t omputation, the Eu lidean urvature and the aÆne urvature of the normalized aÆne CSF satisfy e k

=

and



 e

Le ek

(4.14)

2

 e L 1 4 (4.15)  e =  3 eee + 3 e 2 e ; respe tively. Sin e e and e Le=2 tend to 1 and 0, respe tively, as  ! 1, for ea h large  0 , e(
;  ) is very lose to an ellipse in some time interval ( 0 ;  00 ). By applying a spe ial aÆne transformation to make e (
;  0 ) almost ir ular, we may assume that e, its aÆne ar length, is very lose to 1 in ( 0 ;  00 ). Here  00  0 an be arbitrarily large as  0 tends to 1. Moreover, from the formula

( + ve) = e ; ve = e

ve3 ve

1=2

;

we see that e and e are very small in this interval. For a small "0 to be hosen later, let ( 0;  00) be the largest open interval on whi h 1

© 2001 by Chapman & Hall/CRC

"0

6  6 1 + "0

112

CH. 4.

The Convex Generalized Curve Shortening Flow

and  e

e ; 



6 "0 :

(4.16)

We may ompare (4.15) with the heat equation 1 u = u ; 3 whi h satis es os u = O(exp(  =3)). For some xed small "0 , os  e 6 Ce

1 6

;

8 2 ( 0 ;  00) :

Substituting this estimate into (4.14) and then integrating the inequality, we have, for 2 > 1 in [ 0 ;  00 ),

log ke(; 2 ) 6 C 0e k (; 1 ) e



1 6 1

:

(4.17)

Taking 1 =  0 , we on lude that ek(; 2 ) is arbitrarily lose to 1. Hen e, (4.16) holds for all  >  0 and so  00 = 1. Now (4.17) implies that e (
;  ) onverges uniformly to some 1 as  ! 1. By Lemma 4.16, 1 must be an ellipse. Finally, following the end of the proof in Theorem 4.9, we know that 1 is entered at the origin. The proof of Theorem 4.13 is ompleted. 4.4

Uniqueness of self-similar solutions

In this se tion, we shall prove the following uniqueness result. Under the assumptions  > 1 and ( + ) = (), the equation k = h has a unique solution whi h satis es h( + ) = h( ). Theorem 4.17



To show that the solution exists, we solve (4.1) with a entrally symmetri 0 . By Theorem 3.12(i) (for  = 1) and the theorem in Remark 4.10, we know that its normalized ow sub onverges to a Proof:

© 2001 by Chapman & Hall/CRC

4.4.

113

Uniqueness of self-similar solutions

onvex urve whi h satis es (4.4). After a suitable s aling, we an take  = 1. The uniqueness of this equation is ontained in the following theorem.  Let ' be a ontra ting self-similar solution of the

ow (4.1) satisfying (4.4) with  = 1,  > 1, and symmetri . Then, the relative isoperimetri ratio Theorem 4.18

Z

1 2

Z

hA['℄d


2

hA[h℄d

2

 LA'

is stri tly de reasing along any solution (
; t) of (4.1) (here h is the support fun tion of (
; t)), unless h = ' + h(x0 ; y0 ); ( os ; sin )i for some positive  and (x0 ; y0 ). Proof:

We have 

d L2' dt A

= =





d A210 dt A0





Z

2 AA102 A0 0



'A['℄

 A['℄  d A[h℄

Z



A10

'A['℄



A['℄ d A[h℄

By the Holder inequality, Z





A['℄  d 'A['℄ A[h℄   Z A['℄  1 'A['℄ d A[h℄

Z

>

'A['℄

A1

and so it suÆ es to prove the theorem for  = 1:

© 2001 by Chapman & Hall/CRC



;

 A['℄  A(h)

1



d :

114

CH. 4.



Z

'A['℄

The Convex Generalized Curve Shortening Flow



A['℄ A1 A10 : d > A[h℄ A0

(4.18)

By the H older inequality again,

Z Z

'A['℄

A['℄ d > A[h℄

hA['℄d Z

2

:

h2 A[h℄ d '

So (4.14) follows from

> A1

A10 A0

Z

h2 A[h℄d : '

Let's assume for this moment

h(; t).

Then we an take



h

is symmetri , i.e.,

h='

=

h( + ; t) =

in the anisotropi Bonnesen

inequality Theorem 4.2,

h
2 A1 '

2

h
A01 + A0 '

Multiplying this inequality by

'A[h℄

6

0

:

and then integrating over

S1,

we have

Z

A1

h2 A[h℄d '

A01 A0 + A0 A10

2

6

0

;

whi h is pre isely the desired estimate. Furthermore, we know that

h = ' + h(x0 ; y0 ); ( os ; sin  )i for some

equality holds if and only if positive



x0 ; y0 ).

and (

To prove (4.14) for non-symmetri urves, we pro eed as follows. Mark a point

X ( )

on the urve

and let

Y ( )

 +  ); sin( +  )).

whose outer normal is ( os( ne ting

X

and

D1 and D2 . of

Y

As

be the point on

The line segment on-

divides the region en losed by

into two subregions

X ( ) transverses along the urve

until the positions

X ( ) and Y ( ) are ex hanged, the subregions hange from D1

© 2001 by Chapman & Hall/CRC

and

4.4.

115

Uniqueness of self-similar solutions

D2 to D2 and D1 . By the mean-value theorem, there exists some

X (0 ) su h that jD1 j = jD2 j. The line segment X (0 )Y (0 ) divides

into two ar s + and . We an extend 1 and 2 separately to get

two losed onvex urves + and whi h are entrally symmetri with respe t to the mid-point of X (0 )Y (0 ). Now (4.14) holds for + and . (Noti e that + and are C 1- urves in general. However, Theorem 4.2 ontinues to hold.) Therefore, in obvious notation,

Z

'A['℄

A['℄ d A[h℄

A1

1 = 2



Z

A10 A0

A['℄ d A[h+ ℄ A1

'A['℄

60:

+

A10 A+ 0

1 + 2



Z

A['℄ d A[h ℄ A1

'A['℄

A10 A0





Notes The results in Se tions 1,2 and 4 are largely taken from Andrews [10℄, to whi h we refer for a systemati and rather omplete dis ussion on the AGCSF. Proofs of the results on onvex bodies stated in Se tion 1 an be found in S hneider [99℄. Spe i ally, see x6.6 for the Minkowski inequality, (6.6.10) for Theorem 4.3, and (6.2.21) for the Bonnesen-type inequality. The Blas hke-Santalo inequality, whi h was due to Blas hke in the planar ase, was proved in Blas hke [21℄. (See also [2℄.) In the proof of sub onvergen e, one frequently uses the monotoni ity of some fun tionals su h as the entropy and the \Firey fun tional" F . A ni e dis ussion on these quantities an be found in Andrews [9℄. The monotoni ity of the relative isoperimetri

© 2001 by Chapman & Hall/CRC

116

CH. 4.

The Convex Generalized Curve Shortening Flow

ratio along the ow, Theorem 4.18, was rst proved by Gage [54℄ for the CSF. Uniqueness of the self-similar solution for the symmetri , anisotropi CSF was rst pointed out in Gage [57℄, where it is dedu ed from a lassi al theorem of Wul. Another proof an be found in Dohmen-Giga [44℄. Theorems 4.17 and 4.18 are taken from [10℄. Most ows studied in this book are shrinking. We point out expanding isotropi and anisotropi ows for urves have been studied by several authors, see Urbas [108℄ [109℄, Gerhardt [60℄, Chow-Tsai [38℄, and Andrews [10℄.

Expanding Flows.

One onsequen e of the sub onvergen e of the normalized ow of (4.1) is the existen e of

ontra ting self-similar solutions. A

ording to Theorem 4.9 and Remark 4.10, they exist for  2 (1=3; 1). Moreover, they are unique when  is symmetri and  > 1. This issue an also be studied by looking at the ellipti equation (4.5) dire tly. In fa t, using the degree-theoreti method, Dohmen, Giga, and Mizogu hi [45℄ proved the existen e of these solutions for  > 1=2. In Ai-Chou-Wei [2℄, we use the variational method to prove existen e for  > 1=3. Moreover, similarity is drawn between the Nirenberg problem and the equation (4.5) when  = 1=3. By observing that the spe ial aÆne group SL(2; R ) a ts on S 1 like the onformal group a ts on S 2 in the Nirenberg problem (Chang-Gursky-Yang [26℄), ne essary onditions and suÆ ient onditions for the existen e of self-similar solutions were found. For instan e, a ne essary ondition is the \Kazdan-Warner

ondition", Self-similar solutions for the AGCSF.

Z S1

a0 ( )ei d h2 ( )

© 2001 by Chapman & Hall/CRC

=0:

4.4.

117

Uniqueness of self-similar solutions

In parti ular, when a = 1 + " os ; j"j < 1, (4.1)1=3 does not have any self-similar solution. On the other hand, we have: Theorem.

Let

a( ) = a0 +

X( N

an os 2n + bn sin 2n ) > 0 :

n=1

Suppose that

X

N 1 n=1

j j

j j

n n < N N ;

Then (4.5) has at least (N

1)

n

= an + ibn :

many solutions.

When  2 (0; 1), it is pointed out in [10℄ that self-similar solutions are, in general, not unique. In Chou-Zhang [33℄, we show that stable self-similar solutions are unique for a lass of anisotropi fa tors.

The aÆne urve shortening problem. Our dis ussion on the aÆne CSF basi ally follows Andrews [8℄, where a hypersurfa e driven by its aÆne normal is studied. The problem was also studied in Sapiro-Tannenbaum [98℄. From a ompletely dierent ontext, the

ow arises from image pro essing and omputer vision, where people sear h for geometri ows whi h preserve ellipses. In AlvarezGui hard-Lions-Morel [6℄, the ow, termed as the fundamental equations in image pro essing, was derived by the axiomati method. On the other hand, it was proposed by Sapiro-Tannenbaum [97℄ in the

ontext of aÆne geometry. A general result in [94℄ hara terizes the aÆne CSF as the aÆne invariant urvature ow of the lowest order.

Curve ows in Klein geometry. As a further interesting development of the aÆne CSF, Olver-Sapiro-Tannenbum [94℄ propose the

© 2001 by Chapman & Hall/CRC

118

CH. 4.

The Convex Generalized Curve Shortening Flow

study of invariant urve ows in a Klein geometry (Guggenheimer [70℄ and Olver [93℄). A

ording to the Erlangen Programme, ea h Lie transformation group G a ting on the plane gives rise to a geometry. For a given Lie group or a Lie algebra g (realized as ve tor elds on R2 ), we an de ne its group length element d to be the g-invariant one-form of lowest order, and the group urvature  to be the absolute dierential invariant of lowest order. It is known that all dierential invariants of g are fun tions of  and its derivative with respe t to the group length. Next, one de ne, the group tangent and group normal of a urve to be J =  and N =  , respe tively. Now the \group urve shortening ow" is  =N : t

(4.19)

Noti e that, when the group is the Eu lidean groupnspe ial aÆne group, the ow is the CSFnaÆne CSF. So (4.19) is a generalization of the urve shortening ow to other geometries. The expression for the group urvature for the similarity group, the fully aÆne group, and the proje tion group an be found in [94℄. The reader should be aware that, for some groups, su h as the similarity group, the ow is not urve shortening and also sometimes one needs to modify the equation to retain paraboli ity. As one an see from the variation formula (4.10) for the aÆne length, the ACSF is not pointwise shortening (it is aÆne length shortening, though [98℄). To get a more natural \ urve shortening ow" from the variational point of view, one should look at the L2 -gradient

ow,  = t

N ;

(4.20)

where we have put a minus sign before the aÆne urvature to guarantee paraboli ity. This is a fourth-order paraboli equation. The

© 2001 by Chapman & Hall/CRC

4.4.

Uniqueness of self-similar solutions

119

aÆne length in reases pointwisely along this ow. Noti e that the isoperimetri ratio attains its minima at ir les but the aÆne isoperimetri ratio attains its maxima at ellipses. It is proved in Andrews [11℄ that this urve lengthening ow expands to in nity and tends to an ellipse after suitable normalization.

© 2001 by Chapman & Hall/CRC

Chapter 5 The Non- onvex Curve Shortening Flow In this hapter we rst prove the Grayson onvexity theorem. Theorem 5.1 (Grayson onvexity theorem) where

0

su h that

Consider the CSF

is a smooth, embedded losed urve. There exists a

(
; t)

is uniformly onvex for all

t0 < !

t 2 [t0 ; ! ).

The proof of this theorem is based on the monoton ity of an isoperimetri ratio introdu ed by Hamilton (Se tion 1) and a blowup argument (Se tion 2). Next, we dis uss the lassi ation of the singularities of the CSF for immersed urves in Se tion 3. 5.1

An isoperimetri ratio

Let be a smooth, embedded losed urve and D the region it en loses. We an always nd some line segment in D whose endpoints tou h . It divides D into two subsets D1 and D2 . When does not tou h away from its endpoints, both D1 and D2 are regions. In any ase, we an asso iate to ea h triple ( ; D1 ; D2 ) a number G(

) = L2

1

A1

+

121 © 2001 by Chapman & Hall/CRC

1

A2




;

122

CH. 5.

The Non- onvex Curve Shortening Flow

where L is the length of and Ai is the area of Di (i = 1; 2). We de ne g = g( ) to be the in mum of G( ) over all these admissible triples. It is not hard to see that g( ) is always attained. Let ( 0 ; D1 ; D2 ) T be a minimum of g( ). We rst assert that 0 only onsists of two

omponents ontaining the two endpoints separately. For, if there is T a point in 0 lying outside these two omponents, it divides into two line segments 1 and 2 . We onsider the two admissible triples S S S ( 1 ; D11 ; D2 D12 ) and ( 2 ; D12 ; D2 D11 ), where D11 D12 = D1 and i  D1i (i = 1; 2). We have g ( ) = G(

0

) 6 G( i ) ; i = 1; 2 :

From the de nition of G we have, in obvious notation, A11 (A2 + A12 ) A2 (A11 + A12 )

6 (L +L1L )2 1 2

A12 (A2 + A11 ) A2 (A11 + A12 )

6 (L +L2L )2 1 2

2

and 2

:

Adding up these inequalities yields a ontradi tion: 1+

2A11 A12 A2 (A11 + A12 )

6 1 (L2L+1 LL2 )2 1 2

T

:

Hen e, there are exa tly two omponents in 0 . Next, 0 must be transversal to at the endpoints. For, if not, we an easily nd a variation of 0 in whi h the length de reases at an in nite rate while the area hanges at a nite rate. In summary, T any minimum 0 of g( ) must interse t transversally, and 0

onsists of the two endpoints of 0 . Now we ompute the rst and se ond variations at a minimum 0 . Without loss of generality, we may assume 0 is a verti al line

© 2001 by Chapman & Hall/CRC

5.1.

123

An isoperimetri ratio

segment over x = x0 . Near its top (resp. its bottom) is represented as the graph of a fun tion y = y+ (x) (resp. y = y (x)). For any xed real numbers a and b, onsider the line segment  whi h onne ts (x0 + a; y (x0 + a)) and (x0 + b; y+ (x0 + b) for small . The square of the length of  and the area of the region lying on the left of  are given by L2 () = (b

a)2 2 + y + (x0 + b)


(x0 + a) 2

y

and A1 () = A1 (0) +

x0Z+b

y + (x)dx

x0

x0Z+a

y

(x)dx

x0


1 + y (x0 + b) + y (x0 + a) (b a) ; 2

respe tively. We have dL d

= byx+ (x0 ) ayx (x0 ) ;

d2 L d2

=

dA1 d

=

1

L

i

+ (x0 ) a2 y (x0 )
; (b a)2 + L b2 yxx xx

b+a

2

y + (x0 )

y




(x 0 ) ;

and d2 A1 d2


= ab yx+ (x0 ) yx (x0 ) ;

at  = 0. Now, as the fun tion G(  ) attains minimum at  = 0, we have

© 2001 by Chapman & Hall/CRC

124

0

CH. 5.

=

=

The Non- onvex Curve Shortening Flow

 1 d + log L2 () d =0 A1 () A 2

L

Taking

byx+ (x0 )

A1 ()

b + a




ayx (x0 )

2



1

1

A1

1

A



A1

y+ (x0 )




y (x0 ) :

a = b = 1 we dedu e yx+ (x0 ) = yx (x0 ) :

(5.1)

Putting this ba k to the above expression, we have

yx+ (x0 ) =

L2  4

1

A1

A

1



A1

:

(5.2)

Next we ompute

0

 1 d2 log L2 () + d2 =0 A1 () A

6

2

h

+ (x 0 ) a)2 + L b2 yxx

(b

L2

=



by+ (x ) L2 x 0 2

 1

ab +

Taking

a

A1

h 1

=

A21

b

+

A (A

ayx (x0 ) 1

1



A1

A1 )2



1

A1 ()


a2 yxx(x0 )

i

2

yx+ (x0 )

i b + a 2  2




yx (x0 ) y+ (x0 )

y (x 0 )

2

:

= 1 and using (5.1) and (5.2) in this expression, we

© 2001 by Chapman & Hall/CRC

5.1.

125

An isoperimetri ratio

have 0

6 L2

+ (x0 ) yxx



(A A1 )

yx+ (x0 )




yx (x0 )

h

+L2 =



1

i 1 1 + (x0 ) y (x0 )
2 y + A21 (A A1 )2  1
2 + 1 2 L2 yxx (x0 ) yxx (x0 ) L A1 A A1

+

6

1 A1


2 2 + yx (x0 ) yx (x0 ) 2 L




yxx (x0 )

h

i 1 1 + A21 (A A1 )2


2L2 2 + ; yxx (x0 ) yxx (x0 ) + L A1 (A A1 )

i.e.,
1 + L2 : yxx (x0 ) yxx (x0 ) 6 L A1 (A A1 )

Proposition 5.2 Let

(
; t)

be a solution of the CSF where

losed and embedded. For any than or equal to



(5.3)

t0

2 (0; !)

, either

or it is stri tly in reasing in

g ( (
; t))

(
; t)

is

is greater

(0; t0 ).

For a xed t0 we let (t0 ) be a minimum of g( (
; t0 )). As before, we may assume that (t0 ) is a verti al line segment over x = x0 . By the prior dis ussion, (
; t) is lo ally des ribed by the fun tions y (x; t) for t lose to t0 . Let (t) be the verti al line segment onne ting (x0 ; y (x0 ; t)) and (x0 ; y+ (x0 ; t)). It divides the region en losed by (
; t) into two regions D1 (t) and D2 (t) whi h
lie on its left and right, respe tively. (t); D1 (t); D2 (t) forms an admissible triple for (
; t). Proof:

© 2001 by Chapman & Hall/CRC

126

CH. 5.

The Non- onvex Curve Shortening Flow

The length of (t); L(t), is given by y+ (x0 ; t) y (x0 ; t). Re all that y = y satis es yt = (tan 1 yx)x (5.4) (see (1.3)). Let A1 (t) be the area of D1 (t). It follows from (5.4) that dA1 =  + tan 1 yx+(x0 ; t) dt

tan 1 yx (x0 ; t) :

(5.5)

By (5.1), (5.4), and (5.5) we have  1 1  d + log L2 dt t=t0 A1 A A1

=



2 + yxx(x0 ; t0 ) yxx(x0 ; t0 ) 1 + yx+ (x0 ; t0 )2 1

L



+ =

A

1

1

A1


 + 2 tan 1 yx+ (x0 ; t0 ) + 2

A



A

1

1 A1

A1



2 + yxx(x0 ; t0 ) yxx(x0 ; t0 ) 1 + yx+ (x0 ; t0 )2 1

L

 1 2 tan 1 yx+ (x0 ; t0 ) A1

A

1



A1

+

h A2 + (A

1

A1 (A

A 1 )2 i : A1 )A

By (5.2) and (5.3), we have  1 d 1  + log L2 dt t=t0 A1 A A1

>

2L2

A1 (A

+

>

A1 )

h A2 + (A

1

A 1 (A

2L2

A1 (A

x

 1 + 2yx (x0 ; t0 ) A1

A

1



A1

A1 )2 i A 1 )A

A1 )

© 2001 by Chapman & Hall/CRC


1 + y+ (x0 ; t0 )2 1

L2  1 2 A1

A

1

2

A1

+

h A2 + (A

1

A 1 (A

A1 )2 i A1 )A

5.1.

127

An isoperimetri ratio

=

> =

L2 

1

2 A1

L2 

1

2 A1

+ +

A A

1 1

2

A1 2

A1

+ +

A 1 (A

A1 )2 i A1 )A

1

1

h A2 + (A



1

2 A1

+

A



A1


 1  1 1  G (
; t0 ) : + 2 A1 A A1

When G( (
; t0 )) < , this inequality shows that g( (
; t)) is stri tly in reasing in [t0 Æ; t0 ℄ for some Æ > 0. By repeating the same argument to any minimum of g( (
; t0 Æ)), we an tra e ba k in time and show that g( (
; t)) is stri tly in reasing in (0; t0 ℄.  We shall need a similar isoperimetri ratio where the line segments lie in the exterior of . Let's assume that is an embedded,

losed urve whi h is non- onvex and ontained in the unit disk. Let C be the ir le fx2 + y 2 = 16g. We may onsider the olle tion of all line segments whi h belong to the region D bounded between C and

and whose endpoints tou h . As before, ea h su h line segment divides D into D1 and D2 , and we may form the triple ( ; D1 ; D2 ). The isoperimetri quantity G0 ( ) and g0 ( ) an be de ned in a similar way. It is still true that any minimizing interse ts transversally at the endpoints and its interior belongs to D. Furthermore, the rst and se ond variation formulas (5.1){(5.3) hold for . Let be a solution of the CSF where (
; t) is embedded, losed, and non- onvex for all t. Then, for any xed t0 2 (0; !), either g0 ( (
; t0 )) > =2 or g0 ( (
; t)) is stri tly in reasing in (0; t0 ℄. Proposition 5.3

As in the proof of Proposition 5.2, we assume the minimum of g( (
; t0 )), (t0 ), is verti al over x = x0 , and (
; t) is des ribed by y  (
; t) near the endpoints. Let A(t) be the area of the region D (t)

Proof:

© 2001 by Chapman & Hall/CRC

128

CH. 5.

The Non- onvex Curve Shortening Flow

bounded by (
; t) and C , and A1 (t) the area of the sub-region D1 (t) lying on the left of (t). We have, by (1.19), dA dt

= 2 ;

(5.6)

and (5.5) holds. By (5.1){(5.3), (5.5), and (5.6) we have 

d 1 log L2 dt t=t0 A1

=

A1

L

1

A

A1 (A 

L2

2

1 A1

A1

2L2

+2 =

A



2 + (yxx (x0 ; t0 ) yxx (x0 ; t0 ))(1 + yx+ (x0 ; t0 )2 ) 1 +



>

+

1

1

A 



L2



2

1



A

A1 ) A



1  + 2 tan 1 yx+ (x0 ; t0 ) + 1 A1

1

A



2

+

A1



1

A

A

A

1

1

dA A1 dt



A1



A1

2 (A A1 )2 3A21 1 1 + + : A1 A A1 A1 (A A1 )A

Sin e (
; t) is ontained inside the unit disk, A1 (t) <  and A(t) A1 (t) > 15 . Therefore, 

(A A1 )2 3A2 A1 (A A1 )A

whi h implies 

>

© 2001 by Chapman & Hall/CRC



+

A

1 1 1 + 2 A1 A A1





1 1 + ; 4 A1 A A1



> d 1 log L2 dt A1

2

2

> 2 AA1 +(A(A AA)1A) 1 1

1





A1 

2



g 0 ( (
; t0 )) :





5.2. Limits of the res aled ow

129

Now we an nish the proof of the proposition following the last step in the proof of Proposition 5.2. 

5.2 Limits of the res aled ow By (1.18) and Proposition 1.2, the urvature of the maximal solution of the CSF for an immersed losed 0 be omes unbounded as t " !. Consequently, we an nd a sequen e f (pn ; tn )g, tn " ! and pn 2 S 1 su h that

jk(p; t)j 6 jk(pn ; tn)j;

8 (p; t) 2 S 1  [0; tn ℄ :

We all su h a sequen e an essential blow-up sequen e. Noti e that, by (1.17), the urvature may tend to negative in nity. We res ale the ow by setting

n (p; t) =

(pn + p; tn + "2n t) "n

(pn ; tn )

;

for t 2 [ tn ="2n ; (! tn )="2n ), where "n = jk(pn ; tn )j 1 . So for ea h n, n solves the urve shortening ow, n (0; 0) = (0; 0), and its urvature satis es jkn j 6 jkn (0; 0)j = 1 for t 2 [ tn ="2n ; 0℄. From now on, we shall assume ea h n is parametrized by ar -length. Sin e the limit urve may not be losed, it is onvenient to assume ea h

n is de ned on the real line as a periodi map. By Remark 1.3 and the As oli-Arzela theorem, we an extra t a subsequen e of n whi h onverges to a solution of the CSF in every ompa t subset of R  ( 1; 0℄. The limit solution 1 , de ned in R  ( 1; 0℄, is either losed, or unbounded and omplete for ea h t. Moreover, jk1(p; t)j 6 jk1 (0; 0)j = 1. We have: Proposition 5.4 (i)

(ii) if

1

equal to

1 (
; t)

is uniformly onvex in

(

1; 0℄, and

is embedded and unbounded, its total urvature must be

.

© 2001 by Chapman & Hall/CRC

130

CH. 5.

The Non- onvex Curve Shortening Flow

The total absolute urvature of any CSF for losed

urves is non-in reasing in time.

Lemma 5.5

Proof:

In ea h positive ", we have d dt

Z

("2 + k2 (
; t)) 2 ds 1

(
;t)

Z

=

(
;t)

k (" + k ) 2

2

1 2



Z

=

"2 ("2 + k 2 )

(
;t)

6



Z

2k + k3 ds s2 3 2



k s

2

1

(
;t)

k 2 ("2 + k 2 ) 2 ds

Z

ds

(
;t)

"2 k 2 ("2 + k 2 )

1 2 ds

0:

Letting " # 0, we obtain the desired result. Lemma 5.6 Proof:

Any in e tion point of 1 (
; t) must be degenerate.

As above, we have

d dt

Z

Z

=



1

n (
;t)

n (
;t)

("2 + kn2 (
; t)) 2 ds

" (" + kn ) 2

2

2

3 2



(5.7)

kn s

2

Z

ds

n (
;t)

"2 kn2 ("2 + kn2 )

1 2 ds:

Suppose at some t0 , 1 (s0 ; t0 ) is a nondegenerate in e tion point. As n tends to 1 smoothly on every ompa t set, for suÆ iently large n and small Æ > 0, the following holds: for ea h t 2 [t0 Æ; t0 ℄ and large n, there exists sn (t) near s0 su h that kn (sn (t); t) = 0

and kn

s

© 2001 by Chapman & Hall/CRC

(sn (t); t)



>





1 k1 (s0 ; t0 ) > 0: 2 s

5.2. Limits of the res aled ow

131

Now, the rst term on the right-hand side of (5.7) an be estimated as follows: Z

" (" + kn ) 2

n Z

6

2

sn (t)+"

6

Z

1

3 2



kn s

2

C1 "2 ["2 + C2 (s

sn (t) "

C0

2

C20 "

"2 ("2 + s2 )

C20 "

ds sn (t))2 ℄

3 2 ds

3 2 ds

2C10 C20 1 ; (1 + C20 2 ) 2

=

where the positive onstants C1 ; C2 ; C10 , and C20 depend on 1 (s0 ; t0 ) s only. By integrating (5.7) from t0 Æ to t0 , we have k

Z

n (
;t0 )

(" + kn ) 2

2

1 2 ds

Z

6

1

n (
;t0 Æ)

("2 + kn2 ) 2 ds

2C10 C20 Æ 1 : (1 + C20 2 ) 2

Letting " # 0, we arrive at Z

n (
;tn )

jkn jds 6

Z

n (
;t0 Æ)

jknjds

2C10 C20 Æ 1 : (1 + C20 2 ) 2

Ba k to the uns aled ow, it means that Z

(
;tn +"2n t0 )

jkjds

Z

(
;tn +"2n (t0 Æ))

jkjds 6

2C10 C20 Æ 1 : (1 + C20 2 ) 2

However, sin e the total absolute urvature of is non-in reasing in t, the left-hand side of this inequality tends to zero as n ! 1,

© 2001 by Chapman & Hall/CRC



132

CH. 5.

The Non- onvex Curve Shortening Flow

and the ontradi tion holds. We on lude that non-degenerate in e tion point.

1

annot have any



(i) By Lemma 5.6, does not have any non-degenerate in e tion points. Were it not uniformly onvex, we an nd some t; p, and q su h that k (p; t)k (q; t) < 0 and there is a point r between p and q satisfying k (r; t) = 0. By applying the Sturm os illation theorem to the equation satis ed by k , we infer that, for t > t, t t small, has a non-degenerate in e tion point. This is in on i t with Lemma 5.6. Hen e, (
; t) must be uniformly onvex for all t. (ii) Sin e now is unbounded and embedded, its total urvature is less than or equal to . Let's show that the former is impossible. To prove this, we introdu e oordinates so that, for a xed t0 , (
; t0 ) is the graph of a onvex fun tion U (x; t0 ), x 2 R, with jUx j ! a as jxj ! 1. By omparing (
; t) with verti al lines whi h are stationary solutions of the CSF, every (
; t) is the graph of a smooth fun tion U (x; t). Next, by omparing (
; t) with grim reapers whi h move steadily upward or downward, we infer that the speed of (
; t) is bounded by a onstant. In parti ular, it means that jUx (x; t)j ! a as jxj ! 1 for all t. By dierentiating (5.4), we see that the fun tion w = Ux is a positive solution to the following uniformly paraboli equation in divergen e form,

Proof of Proposition 5.4

1

1

1

1

1

0

0

1

1

1

wt =



wx 1 + w2



; x

and hen e it is onstant by Moser's Harna k inequality (Theorem 6.28 in [88℄). It means (
; t) is a horizontal line for all t, ontradi ting with its uniform onvexity. So, the total urvature of is always equal to . The proof of Proposition 5.4 is ompleted. Now, we an prove the Grayson onvexity theorem. If is bounded, it follows from Proposition 5.4 that, for suÆ iently large 1

1

1

© 2001 by Chapman & Hall/CRC

5.2. Limits of the res aled ow

133

is uniformly onvex too. S aling ba k, we on lude that (
; t) is uniformly onvex for all t lose to !. If 1 is unbounded, by Proposition 5.4, it an be expressed as the graph of a family of onvex fun tions U (x; t), x 2 R or ( a; a), satisfying Ux (x; t) ! 1 as x ! 1 (or a). We shall show that this will lead to a ontradi tion. For any " > 0, we an nd x" su h that 1 U (x; 0) > x" ; x 2 ( a; a); jxj > x" ; n, n

"

where we have assumed U (0; 0) = 0. Consider the horizontal line segment = fy = 1" x"g\fy > U (x; 0)g. It is lear that the length L of is not greater than 2x" ; and, by onvexity, the area A bounded between and 1(
; 0) is not less than 2 1 L
x"=". For suÆ iently large n, the length Ln of n = fy = x"="g \ fy > n (
; 0)g and the area An of the region Dn0 bounded between n and n (
; 0) are arbitrarily lose to L and A, respe tively. Therefore, when n belongs to Dn (the region en losed by n (
; 0)), the triple ( n ; Dn0 ; Dn n Dn0 ) is admissible, and we have g ( n (
; 0))

6 G( n) < 2L2

 1 1 2L


x""

+

1

Cn



;

where Cn tends to 1 as n ! 1. By the Eu lidean and s aling invarian e of g( ), it shows that g( (
; t)) ould be arbitrarily small for some t lose to !. But this is impossible by Proposition 5.2. On the other hand, if n belongs to the exterior of n (
; 0), we an then use Proposition 5.3 instead of Proposition 5.2 to draw the same

ontradi tion. (Without loss of generality, we may assume (
; t) is

ontained in the unit disk.) Hen e, 1 must be unbounded. We have nished the proof of the theorem.

© 2001 by Chapman & Hall/CRC

134

CH. 5.

The Non- onvex Curve Shortening Flow

5.3 Classi ation of singularities First of all, we note that the blow-up rate of the urvature for the CSF has a universal lower bound. For, it follows from (1.17)0 that dkmax dt

3 6 kmax :

By integrating this inequality from t to !, we obtain 

kmax (t) > 2(!

t)

 1=2

:

With a similar estimate on kmin(t), we on lude

jkjmax(t) >



2(!

t)

 1=2

(5.8)

:

Let (
; t) be a CSF for losed, immersed urves. By Theorem 6.4, there are nitely many singularities as t approa hes !. A point Q in R2 in alled a singularity for the CSF (
; t) if there exists f(pj ; tj )g, pj 2 S 1 , tj " !, su h that (pj ; tj ) ! Q and jkj(pj ; tj ) ! 1 as j ! 1. The singularity is of type I if there exist a onstant C and a neighborhood U of Q whi h is disjoint from other singularities su h that p

t) 6 C

sup jkj(
; t) 2(!

U

(5.9)

for all t 2 [0; !). It is of type II if (5.9) does not hold. When Q is a type I{singularity, it is natural to res ale it by setting 

e(
;  ) = 2(!

 1=2

t)

(
; t)




Q ;

where 2 = log(! t). When (
; t) is embedded, this normalization oin ides with the one we used in Chapter 3. Noti e from (5.8), (5.9), and paraboli regularity, that the urvature of e is uniformly pin hed between two positive onstants and its derivatives are uniformly bounded for all  .

© 2001 by Chapman & Hall/CRC

5.3. Classi ation of singularities Theorem 5.7 Let

(
; t)

135

be a CSF for losed, immersed urves.

Suppose that all singularities are of type I. Then, ea h sequen e





e(
; j )

ontains a subsequen e onverging to a self-similar solu-

tion of the CSF ontra ting to the origin as

j

!1

.

It follows from this theorem the ow eventually be omes onvex and shrinks to a point. The proof of Theorem 5.7 is based on a monotoni ity formula. Let the \normalized ba kward heat kernel" be 2 (X ) = e jX j =2 ; X

We de ne

M ( ) =

Z

2R

2

:




 e(
;  ) dse ;

e(
; )

where se is the ar -length parameter of e. Lemma 5.8

d M( ) = d Proof:

Z

e(
; )



2

 e k + h e; ni dse :

e and its ar -length se satisfy the equations

et = e k n + e ;

and se = ( e k 2 + 1)se :

Therefore, d M ( ) = d

= =

© 2001 by Chapman & Hall/CRC

Z

(

Z Z

e k 2 + 1)

 je k n + ej2

Z

h e; e kn + e

idse


h e; esei dse

je k + h e; nij2 dse;

136

CH. 5.



after a dire t omputation. Now, by (5.9),

j (s; t)

Qj

The Non- onvex Curve Shortening Flow

6 6

Z

!

p

t

jkjdt

C !

t:

This means

j e(s;  )j 6 C ; that is, e is ontained in the disk DC for all  . Sin e all derivatives of ek are under ontrol, it follows from Lemma 5.8 that Z lim je k + h e; nij2 dse = 0 : 

!1

Thus, any sequen e f e(
; tj )g has a subsequen e onverging to a

losed urve  solving k  + h  ; ni = 0 :

By reversing the orientation of  , if ne essary, k is positive somewhere. We may introdu e the support fun tion of  , h , and then this equation be omes h = k  :

So,  is a ontra ting self-similar solution. All self-similar solutions of the CSF have been lassi ed in Chapter 2. Sin e  is losed, it must be one of the Abres h-Langer urves. The proof of Theorem 5.7 is ompleted. Next, onsider a type II singularity Q. Suppose that there is an essential blow-up sequen e onverging to it in some neighborhood of Q, U , whi h ontains no other singularities. It is appropriate to use

© 2001 by Chapman & Hall/CRC

5.3. Classi ation of singularities

137

the normalization des ribed in the previous se tion. Adapting the same notation, the normalized sequen e n sub onverges to a limit

ow 1 (
; t), t 2 ( 1; 0℄. Theorem 5.9 Let

Q

be a type II singularity.

Suppose that there

exists an essential blow-up sequen e onverging to borhood

U . Then 1 is a grim reaper.

Q

in some neigh-

We point out that the assumption of the existen e of an essential blow-up sequen e onverging to Q is not ne essary. In fa t, a

ording to Theorem 6.4, singularities of the limit urves  as t " ! are nite. Hen e, one an apply the blow-up argument in any xed, suÆ iently small neighborhood of ea h singularity, where an essential blow-up sequen e always exists. First of all, by the de nition of a type II singularity and the de nition of n , we know that n sub onverges to 1 in every

ompa t subset of R. Hen e, 1 is a CSF whi h exists for all positive and negative t. (A solution whi h exists for all time is alled an eternal solution.) In the previous se tion, we have shown that it is uniformly onvex. We laim that it has no self-interse tion. For, the rate of de rease in the area of a loop is equal to the dieren e of the normal angles at the meeting point, whi h is at least . Hen e, 1

annot exist for all time if it has a loop. Now, as before, one an show that the total urvature of 1 must be equal to . We write = 1 for simpli ity. We laim that a onvex eternal solution must be a travelling wave. For, let f = kt and g = 2(t t0 )f + k and parameterize the ow by the normal angle. Then, Proof:

ft = k 2 (f + f ) +

© 2001 by Chapman & Hall/CRC

2f 2 k

138

CH. 5.

The Non- onvex Curve Shortening Flow

and 2f


gt = k 2 g + k 2 +

k

g:

When is losed, the strong maximum prin iple implies that g > 0 for all t > t0 . By approximating by a losed, onvex ow, one sees that it ontinues to hold for the unbounded . Letting t0 go ba k to 1, we on lude that kt > 0 for all t 2 R. Now, when is losed, onsider its entropy:

E (t ) =

Z

log k(; t)d :

As before, we have d2 E (t ) = 2 dt2

>

2

Z

kt2 d k2

dE (t)
2 : dt

So  dE (t) 6 dt 2(T t)

for all T > t. By an approximation argument, this inequality still holds for unbounded . Letting T ! 1, we on lude that kt = 0, i.e., k is a travelling wave for the CSF.  Type II singularities may o

ur even when the

ow shrinks to a point. For example, let 0 be a gure-eight whi h is symmetri with respe t to the x-axis and the y-axis with exa tly one in e tion point lo ated at the origin. It is easy to see that the

ow exists until the area en losed by ea h leaf be omes zero. By symmetry, the limit urve is either a point or a line segment. The latter is ex luded by the strong maximum prin iple. So, the ow

Example 5.10

© 2001 by Chapman & Hall/CRC

5.3. Classi ation of singularities

139

shrinks to a point is nite time. Now, sin e there are no Abres hLanger urves with two leaves, the singularity must be of type II. In general, singularities of gure-eight were studied in Grayson [67℄, where it is shown that the dieren e of the en losed area of the leaves is onstant in time. When the areas are equal but the leaves are not symmetri , it is not known whether the ow shrinks to a point or not. In any ase, the singularities are of type II. A ardoid is a losed urve with winding number 2, symmetri with respe t to the x-axis, and on whi h the urvature has one maximum, one minimum, and no other riti al points in between these. It is lear that the inside losed loop will shrink to a point before the outside one does, and, hen e, forms a type II singularity. A deli ate analysis, arried out in Angenent-Velazquez [18℄, shows that Example 5.11

kmax (t) = 1 + o(1)


h log

j log(! !

t

t)j i1=2

and the asymptoti shape near the usp at ! looks like y(x) =


x  + o(1) : 4 log j log xj

Notes

The proof of the Grayson onvexity theorem presented here is different from the original proof and is taken from Hamilton [72℄. This approa h by res aling, blowing up and analyzing the singularities has been used in many problems in geometri analysis, in luding nonlinear heat equations (Giga-Kohn [65℄), harmoni heat ows (Struwe [105℄), Ri

i ows (Hamilton [73℄), and mean urvature ows (Huisken and Sinestrari [79℄). The results in Se tion 5.3 are mainly based on Alts huler [3℄.

© 2001 by Chapman & Hall/CRC

140

CH. 5.

The Non- onvex Curve Shortening Flow

Similar results an be found in Angenent [15℄ in the onvex ase. The monotoni ity property, Lemma 5.8, whi h also holds for the mean urvature ow, is due to Huisken [77℄. For monotoni ity properties in other geometri problems, see [65℄, [105℄, and [73℄. Isoperimetri ratios

In [72℄, Hamilton gave two isoperimetri ratios, both of whi h are non-de reasing along the CSF. We have used the rst one and omplemented it with Proposition 5.3 due to Zhu [113℄. The se ond one, whi h is more pre ise, an be des ribed as follows. Let be any

urve whi h divides the region D en losed by (
; t) into two regions D1 and D2 . Consider the shortest urve whi h divides the disk D , whose area is equal to jDj, into two regions of area equal to jD1 j and jD2 j, respe tively. De ne H ( ) =

length j length j

j j

and h( ) = inf H ( ) :

Then, h( (
; t)) in reases along the ow. More re ently, two more s aling-invariant geometri ratios were introdu ed by Huisken [78℄. Let ` and d be, respe tively, the intrinsi and extrinsi distan e: S 1  S 1  [0; w) ! R on (
; t). De ne d ; `

I ( )

=

J ( )

= d

i( ) =

.L 

sin

inf I ( ) ;

(p;q)

and j ( ) =

inf J ( ) :

(p;q)

© 2001 by Chapman & Hall/CRC

` L

(L is the perimeter of (
; t)) ;

5.3. Classi ation of singularities

141

Then, i( (
; t)) and j ( (
; t)) in rease along the ow. (For i( (
; t)) we need to impose the assumption that the in mum is attained in the interior.) Both h and j an be used in pla e of g to prove the Grayson onvexity theorem. We shall dis uss i( ), whi h is espe ially ee tive for non- losed urves, in some detail in Chapter 8. The ratio i( ) is very useful in the study of various boundary value problems of the CSF ([78℄). Combined with the blow-up argument, it an be used to ontrol the urvature of the ow away from the boundary. For example, the following result is valid: Theorem Let 0 : I ! R be an embedded urve lying between the strip fa1 < x < a2 g with endpoints Pi on the verti al line x = ai , i = 1; 2. Then, the CSF has a unique solution (
; t), 2 [0; 1), satisfying (ai ; t) = Pi ; i = 1; 2. Moreover, (
; t) tends to the line segment onne ting P1 and P2 as t ! 1. See Huisken [76℄, Polden [95℄, and Stahl [102℄ for further results on the boundary value problems of the CSF. Boundary value problems for the CSF.

In E ker-Huisken [47℄, it was proved that the mean urvature ow for any entire graph has a solution for all time. This is rather striking be ause the behaviour of the initial hypersurfa e at in nity does not ae t solvability. In [95℄, long-time existen e is established for any initial urve whose ends are asymptoti to some semi-in nite lines. In Chou-Zhu [34℄, we prove the following general result. The CSF for omplete, non ompa t urves.

Theorem. Let 0 be any embedded urve whi h divides the plane into two regions of in nite area. Then, the CSF has a solution for all time.

© 2001 by Chapman & Hall/CRC

142

CH. 5.

The Non- onvex Curve Shortening Flow

An example has been onstru ted to show that ! is nite when one of the regions has nite area. In the same paper, it is also proved that the solution is unique when the ends of 0 are graphs over some semi-in nite lines. Long-time behaviour, su h as onvergen e to an expanding self-similar solution or a grim reaper, an be found in [46℄ and [95℄. The long-time existen e of the urvature-eikonal ow for any initial entire graph is established in Chou-Kwong [30℄. Motion of urves in spa e.

Flows in spa e may be written as

 = F n + Gt + Hb t

(5.10)

where b is the binormal of (
; t) and F; G and H depend the urvature and the torsion. The CSF (H = G = 0 and F = k) was studied by Alts huler-Grayson [4℄ where long time existen e was established for a spe ial lass of urves alled \ramps". However, the purpose of [4℄ is to use spatial evolution as a mean to study the long time behavior of the CSF in the plane, espe ially after the formation of singularity. A dierent approa h whi h is based on expressing the

ow as a weakly paraboli system was developed in De kelni k [41℄ in Rn (n > 2). For the level-set approa h one an onsult AmbrosioSoner [7℄. It is worthwhile to point out that the general ow (5.10) sometimes arises in appli ations, for example in the dynami s of vortex laments in uid dynami s (F = G = 0 and H = k, Hasimoto [74℄ and [91℄) and in the dynami s of s roll waves in ex itable media (Keener-Tyson [83℄).

© 2001 by Chapman & Hall/CRC

Chapter 6 A Class of Non- onvex Anisotropi Flows In this hapter, we ontinue the study of ows for non- onvex urves. Let  and be two smooth, 2-period fun tions of the tangent angle satisfying () > 0 (6.1) and

( + ) = () ; ( + ) = () :

(6.2)

We onsider the Cau hy problem for  t

= (k + )n ;

(6.3)

where the initial urve 0 is a smooth, embedded losed urve. This

ow may be regarded as the linear ase for the general ow (1.2), where F is uniformly paraboli and symmetri . Remember that the

ondition (6.2) means that F is symmetri . Without this ondition, embeddedness may not be preserved under the ow. When 0 is

onvex, we have shown in x3.2 that the ow also preserves onvexity and it shrinks to a point where ! is nite. In this hapter we shall show that the Grayson onvexity theorem holds for (6.3). Theorem 6.1

Consider the Cau hy problem for (6.3) where (6.1)

and (6.2) hold. For any embedded losed

143 © 2001 by Chapman & Hall/CRC

0 ,

there exists a unique

144

CH. 6.

A Class of Non- onvex Anisotropi Flows

solution to the Cau hy problem in [0; !) where ! is nite. Moreover, for ea h t in [0; !); (
; t) is embedded losed and there exists some t0 su h that (
; t) be ome uniformly onvex in [t0 ; ! ).

Noti e that Theorem 6.1 redu es to the Grayson onvexity theorem when   1 and  0. The method we are going to use to prove this theorem is ompletely dierent from that of the previous hapter. Here we follow the elementary and geometri approa h initiated by Grayson [66℄ and subsequently developed by Angenent [14℄. It is based on the onstru tion of suitable foliations and then uses them to bound the urvature of the ow. We shall see that the Sturm os illation theorem plays a ru ial role in this approa h. 6.1

The de rease in total absolute urvature

Consider the Cau hy problem for (6.3). From now on, we shall always assume (6.1) and (6.2) are in for e and 0 is an embedded losed

urve. A

ording to Propositions 1.2 and 1.4, it admits a unique smooth solution (
; t) in a maximal interval [0; !). When (6.2) holds and 0 is embedded, ea h (
; t) is embedded. First of all, let's show that ! is nite. In fa t, by (1.18), the length of (
; t) satis es dL dt

=

Z

()k2 ds :

By the Holder inequality, L2 (t) 6 L2 (0)

82 mint :

So, ! is bounded above by a onstant depending only on the length of the initial urve. Next, we examine the in e tion points of the ow. By (1.16),

© 2001 by Chapman & Hall/CRC

6.1.

145

The de rease in total absolute urvature

the urvature of ( ; t), k( ; t), satis es the equation


kt




= (k + ) + k2 (k + ) ; ss

where s = s(t) is the ar -length parameter of ( ; t). Using =s = k , this equation an be rewritten in the form


kt

= s

2

kpp + bkp + k ;

where the oeÆ ients b and are smooth. Sin e ( ; t) is a losed

urve and so its urvature never vanishes identi ally, it follows from Fa t 7 in 1.2 that the zero set of k( ; t), i.e., the in e tion points of ( ; t), is nite for every t (0; !). Moreover, the number of in e tion points drops pre isely at those instants t when ( ; t) has a degenerate in e tion point, and all these instants form a dis rete set in (0; !).


x







2




Let's assume that the ow is always non onvex, for otherwise Theorem 6.1 holds trivially and there is nothing to prove. Thus, there

are at least two in e tion points on the solution urve for ea h time. Without loss of generality, we may assume the number of in e tion points on ( ; t) is onstant in (0; !). Denote the set of in e tion points on ( ; t) by S (t). There exist N > 2 and smooth fun tions p1 (t); ; p ( t) to S 1 su h that, for ea h j = 1; ; N , the ar

(t)

( ; t ) (p +1 p1 ) has non-vanishing urvature [ ( ) +1 ( )℄ ex ept at their endpoints. Re all that the tangent angle at (p; t) is determined as follows. First, we assign its value at a xed (p0 ; t0 ). Then, we extend it to all (p; t) by ontinuity. As a result, any two hoi es of tangent angles dier by a onstant multiple of 2 everywhere. Let  (t) be the tangent angle at (p (t); t), and let      (t);  +1 (t) or  +1 (t);  (t) be the range of the tangent angle of a onvex or on ave ar . Clearly, the tangent angle along an ar is stri tly in reasing or de reasing depending on whether the ar is uniformly onvex or on ave.










j








N




pj t ;pj

t

N

j

j

j

© 2001 by Chapman & Hall/CRC



j

j

j

146

CH. 6.

A Class of Non- onvex Anisotropi Flows

We laim that j is stri tly in reasing and j +1 is stri tly de reasing in time along a onvex ar j . This implies that the interval   j (t); j +1 (t) is stri tly nesting in time. To prove the laim, let t0 be any xed time and let (p0 ; t0 ) be in S (t0 ). By representing the solution as a lo al graph (x; u(x; t)) over, say, the x-axis, the fun tion u satis es ut

p

= 1 + u2x ()k + ()


where (p0 ; t0 ) = x0 ; u(x0 ; t0 ) ,  = tan this equation, we have t

1




;

ux .

By dierentiating




= os2  x x + ( os   + sin )x :

Sin e (p0 ; t0 ) is a non-degenerate in e tion point, the tangent angle at (p0 ; t0 ), (p0 ; t0 ) is either a stri t lo al maximum or minimum. For some small Æ > 0, we have either  >  (p0 ; t0 )

or  <  (p0 ; t0 )

on the paraboli boundary of (x0 Æ; x0 + Æ)  (t0 ; t0 + Æ). By the strong maximum prin iple, we on lude that either (x0 (x0

min  (
; t) >  (p0 ; t0 ) or Æ;x +Æ ) 0

max  (
; t) <  (p0 ; t0 ) ; Æ;x +Æ ) 0

for t 2 (t0 ; t0 + Æ). Noti ing that (p0 ; t0 ) is a lo al stri t maximum (resp. minimum) when it is equal to j +1 (resp. j ), our laim follows. By a similar argument, one an show that the range of the tangent angle of a on ave ar is also stri tly nested.

© 2001 by Chapman & Hall/CRC

6.2.

147

The existen e of a limit urve

Finally, we note that the total absolute urvature of (
; t), when it has no degenerate in e tion points, is given by N X  j =1

j +1 (t)



j (t) :

Therefore, it de reases stri tly as long as (
; t) is non onvex. Summing up, we have proved the following proposition. There exists a unique solution to the Cau hy problem for (6.3) in (0; !) where ! is nite. When (
; t) is non- onvex in (0; !), the range of tangent angles of ea h onvexn on ave ar of

(
; t) is stri tly nesting. As a result, the total absolute urvature of

(
; t) is stri tly de reasing. Proposition 6.2

6.2

The existen e of a limit urve

The total absolute urvature of a losed onvex urve is always equal to 2. In view of Proposition 6.2, the ow (6.3) has a tenden y towards onvexity. We shall rst use this fa t to show that a limit

urve exists when t approa hes !. To a hieve this goal, we need to \foliate" the solution urves. By a foliation we mean a smooth dieomorphism F from I  [0; 1℄, where I is either a losed interval or a losed ar in S 1 , to R2 su h that the leaf F (;
) is a smooth urve without self-interse tions for ea h . We shall denote a foliation by F (
) or by F (;
). Given a foliation F , we an solve (6.3) using ea h leaf as the initial urve. Granted solvability, we obtain in this way a family of time-varying foliations F (
; t), t 2 [t1 ; t2 ℄. To make use of the foliation, we a tually need to \bend" it to obtain two foliations whi h lie on its left and right, respe tively. For ea h small Æ > 0, let's postulate the existen e of two foliations F + and F of (6.3) whi h satisfy

© 2001 by Chapman & Hall/CRC

148

CH. 6.

A Class of Non- onvex Anisotropi Flows

(H1 ) jF (p; t) F (p; t)j 6 Æ ; (p; ) 2 I  [0; 1℄ ; t 2 [t1 ; t2 ℄; and (H2 ) Any two dierent leaves from F , F + , and F are either disjoint or meet transversally at exa tly one point. At any interse tion point of a leaf of F + (
; t) and a leaf of F (
; t), there passes a leaf of F (
; t) su h that the leaf of F + (
; t) (resp. the leaf of F (
; t)) lies on its right (resp. left). The angle between the leaves has a uniform positive lower bound. By an evolving ar , we mean a smooth map : ! R2 S where = f[p1 (t); p2 (t)℄  ftg : t 2 [0; T )g and [p1 (t); p2 (t)℄ is a

ontinuously hanging ar of S 1 su h that (
; t) solves (6.3) in [0; T ) and its endpoints are always separated: inft j (p1 (t); t) Proposition 6.3

Let

(
; t); t

(p2 (t); t) j> 0 :

2 [0; T ), be an evolving ar of (6.3).

Suppose that there are three foliations des ribed as above and satisfying (i)

(
; 0)

is transversal to

meet; and (ii) for ea h t, the endpoints

F (
; 0) (
; t)

F  (
; t), and the endpoints 

from

(
; t).

F(
; 0)

whenever they

are disjoint from

of F (
; t) and F (
; t)

F (
; t) and are disjoint

 2 [0; 1℄ and let D be a region ompa tly supported inside Q(t) for all t. Then, there exists a onstant C depending on F , F  , (
; 0); and D so that the urvature of f (p; t) : p 2 \ Dg is bounded by C (provided T D is non-empty).

Let

Q(t)

=

S

and

F (
; t) :

For ea h t 2 [0; T ), F de nes a dieomorphism from [0; 1℄  [a; b℄ to Q(t). By (ii), we may assume ea h (p; t) enters F at F (0; t) and leaves Q(t) through F (1; t). As long as it is transveral to F (
; t), Proof:

© 2001 by Chapman & Hall/CRC

6.2.

149

The existen e of a limit urve

and  are in one-to-one orresponden e, and, by the inverse fun tion theorem, p

F

(; y(; t); t) = (p(; t); t)

for some smooth y. By dierentiating this equation, F 

p + y y = p 

;

2F

2F

F

2F 2

   + 2 y y + 2 y2 + y y y 

= pp



F

p 2 2p + p 2 ;  

and F t

+ y yt = p p + t : t F

It follows that t = k

=

F 



D

+ y y F

.

F 



+ y y ; F

2F

2F

2F

  n; y y + 2 + 2 y y + 2 y2 y F

E.

 F 2 ; 

and

yt = ()

 F 2 y + B (y; y ) ; 

(6.4)

where B depends on , y , and the partial derivatives of (evaluated at (; y; t)). On the other hand, denote the preimages of + ( ; t) and ( ; t) under ( ; t) by b+ ( ; t) and b ( ; t), respe tively. Let Db be a region

ompa tly supported inside (0; 1) (a; b) su h that it ontains all preimages of D under ( ; t), t [0; T ℄. By hoosing Æ suÆ iently F

F

F


F




F






F


© 2001 by Chapman & Hall/CRC

2




F




150

CH. 6.

A Class of Non- onvex Anisotropi Flows

b for ea h t. Moreover, by small in (H1 ), Fb+ (
; t) and Fb (
; t) foliate D (H2 ), ea h leaf in Fb+ (resp. Fb ) is the graph of a stri tly in reasing (resp. stri tly de reasing) fun tion z + (resp. z ) whose gradients are bounded by a onstant C0 . In view of the formula for k, it suÆ es to bound y and y . Sin e  z also satisfy (6.4) and (ii) holds, the Sturm os illation theorem gives C0 6

z 

+

y z 6  6  6 C0 :

In parti ular, it implies that (
; t) is always transversal to F  (
; t) in b Now, as D and so y satis es (6.4) whenever (; y(; t)) belongs to D. in the proof of Proposition 1.9, the fun tion y satis es a uniformly paraboli equation in divergen e form. By Theorem 3.1 in Chapter 5 of [86℄, we on lude that y is bounded in any ompa t subset of b D. 

Suppose (
; t) is a maximal solution of (6.3) in [0; !) where 0 is immersed and losed. Then, as t " !, the urve (
; t)

onverges to a limit set  in the Hausdor metri . The limit set

 is the image of a Lips hitz ontinuous map. When  is not a point, there exist nitely many points fQ1 ;


; Qm g on  su h that

 nfQ1 ;


; Qm g onsists of smooth urves. Away from these points

(
; t) onverge smoothly to  . Theorem 6.4

We begin by reparametrizing the urve (
; t) so that j p (p; t)j, p 2 S 1 , is a onstant depending on t only. We may assume that

(
; t) does not shrink to a point. Then, there are two positive numbers L1 and L2 so that the length of (
; t) is always bounded between L1 and L2 . So, j p (
; t)j is bounded between L1 1 and L2 1 . On the other hand, by Proposition 1.10, we know that the image of

(
; t); t 2 (0; !), is ontained in a bounded set. By the As oli-Arzela Proof:

© 2001 by Chapman & Hall/CRC

6.2.

151

The existen e of a limit urve

theorem, we an extra t a subsequen e for (
; t) whi h onverges uniformly to some Lips hitz ontinuous map  . By Proposition 1.10, it is lear that (
; t) onverges to  in the Hausdor metri , that is, for every " > 0, there exists t0 su h that (
; t) is ontained in the "-neighborhood of  for all t 2 [t0 ; !). The urvature of (p; t), k(p; t), indu es a metri jk(p; t)jdp on the unit ir le. Denote the push-forward of this measure under (
; t) by t . Then f t g is a family of uniformly bounded Borel measures on R2 . We an sele t a subsequen e tn whi h onverges to some Borel measure weakly. This limit measure an be de omposed into atoms and a ontinuous part,

K

K

K

K

K=K K

+

XK Æ j

=1

j Qj

;

where (fP g) = 0 for any point P , fQj g is at most a ountable set of points, arranged in the way K1 > K2 > K3 >


> 0. We let m be the integer for whi h Km >  and Km+1 < . We shall show that fQj g ontains m many points. In fa t, let P be any point in  nfQ1 ;


; Qm g. We an nd , " > 0, and n0 su h that tn (D" (P )) <   for all n > n0 . The preimage of D" (P ) under any one of these n = (
; tn ) is a ountable disjoint union of intervals in S 1 . Sin e the length of (
; t) is uniformly bounded, the number of these intervals whose images also interse t D"=2 (P ) has a nite upper bound. By passing to a subsequen e, if ne essary, we may assume there are exa tly N ar s of n (
) ontained in D" (P ) whi h also interse t D"=2 (P ) for all n > n0 . The ranges of tangent angles of these omponents are intervals. At ea h tn, the sum of the lengths of these intervals does not ex eed  . By rotating the oordinates and passing to a subsequen e, again, if ne essary, we may assume the intervals [=2 =(3N ); =2+ =(3N )℄ and [3=2 (3N ), 3=2 + =(3N )℄ lie in the omplement

K

© 2001 by Chapman & Hall/CRC

152

CH. 6.

A Class of Non- onvex Anisotropi Flows

of the range of tangent angles for D" (P ) \ n (
) for all n > n0 . In other words, the ar s are graphs of some Lips hitz ontinuous fun tions yn;1(x);


; yn;N (x) whose Lips hitz onstants are bounded by

ot =(3N ). Sin e f (
; t)g tends to  in the Hausdor metri , the fun tions fyn;i (x)g onverge to  uniformly as n ! 1. Taking P to be the origin, we an nd ;  > 0,  >  tan =(4N ) su h that the two lines ` = [ ;  ℄  fg lie inside D"=2 (P ) and are disjoint from (
; t) and  for all t > tn0 . A

ording to the basi theory of ODEs, there is a unique stationary solution of (6.3) with (0; ) as endpoints for small ". Denote this solution ar by !0 and translate it along the x-axis to obtain a foliation F onsisting of stationary ar s !x ; x 2 [ ;  ℄. When "0 is suf iently small, we may assume the tangent angle of every leaf lies in [=2 =(400N ); =2 + =(400N )℄. Consequently, for ea h tn > tn0 ,

(
; t) interse ts !x , at most, N times. By passing to a subsequen e again, we may assume the number of interse tion, is always N . Similarly, we may onsider stationary ar s !0+ (resp. !0 ) onne ting (0; ) and ( tan =(100N ); )) (resp. (  tan =(100N ); )). For small ", the ranges of their tangent angles are ontained in [=2 =(100N ) =(400N ), =2 =(100N ) + =(400N )) (resp. [=2 + =(100N ) =(400N ), =2 + =(100N ) + =(400N )℄). Translate them to obtain two foliations F + and F . Now, we an nd a small disk D ontaining the origin that is

overed by F ; F + , and F . By applying Proposition 6.3 to ea h

omponent of (
; t), t > tn0 , and the foliations F  , we on lude that the urvature of (
; t) is uniformly bounded in D near !. Letting t " !, this implies that  annot have a singularity at Q. The

ontradi tion holds. 

© 2001 by Chapman & Hall/CRC

6.3.

153

Shrinking to a point

6.3

Shrinking to a point

In this se tion, we show that the ow (
; t) of (6.3) shrinks to a point as t " !. Let (p; t), p 2 [p1 (t); p2 (t)℄, t 2 (t1; t2 ), be an embedded evolving ar given by (
; t) = (
; t) \ D, where D is a disk, and  (
; t)

onsists of two distin t points on the boundary of D. For any point P on (
; t), let (P; t) be the tangent angle of (
; t) at P . We onsider (t; ) = max

n

(P; t)

(P 0 ; t)

:

P; P 0

o

any points on (
; t)

:

So, (t; ) measures the maximal hange in tangent angles along a

onvex or on ave ar of (
; t). Let (
; t) be an evolving ar of an embedded losed ow of (6.3) des ribed as above. Suppose that (i) the urvature of (
; t) is uniformly bounded in a Æ-neighborhood of the paraboli boundary of S f[p1 (t); p2 (t)℄ftg : t 2 [t1; t2 )g by C0 , and (ii) there exists C1 > 0 su h that Lemma 6.5

sup (t; ) 6 C1 : t

(6.5)

Then, the urvature of (
; t), t 2 [t1 ; t2 ), is uniformly bounded in [p1(t); p2 (t)℄  [t1 ; t2 ) by a onstant depending on C0 and C1.

Before proving this lemma, let's rst show how to use it to dedu e the following result. Let (
; t) be an embedded solution of (6.3). Then, it shrinks to a point as t " !.

Theorem 6.6

Suppose on the ontrary that (
; t) does not shrink to a point. A

ording to Theorem 5.4, we an nd a limit urve  and Proof:

© 2001 by Chapman & Hall/CRC

154

CH. 6.

A Class of Non- onvex Anisotropi Flows

fQ1 ;


; Qmg   ; m > 1, su h that the urvature of (
; t) be omes unbounded near Qj ; j = 1;


; m. For ea h Q = Qj , we an nd a small " > 0 su h that (
; t) = (
; t) \ D"(Q) satis es that hypotheses of Lemma 6.5. Noti e that (t; ) is always uniformly bounded by the total absolute urvature of (
; t), whi h is stri tly de reasing along the ow. By Lemma 6.5, we arrive at a ontradi tory on lusion|the urvature of (
; t) is uniformly bounded. So

(
; t) must shrink to a point.  We shall prove Lemma 6.5 by an indu tion argument on C1 in two steps. Step 1 Assume that the hypotheses of Lemma 6.5 hold and, in addition, C1 6   for some  > 0. We prove the lemma. n  for Step 2 Assuming that Lemma 6.5 holds whenever C1 6 4 some n > 4, we show that it ontinues to hold for all C1 6 n +4 1  . Combining Step 1 and Step 2, it follows from indu tion that Lemma 6.5 holds for any positive onstant C1. Suppose on the ontrary there is an ar (
; t) satisfying the hypotheses of the lemma, but its urvature blows up as t " t2 . Let f (pn ; tn)g be an essential blow-up sequen e inside D (see x5.2). Following the blow-up pro edure there, we an nd a limit urve 1 : ( 1; 0℄  R ! R2 satisfying Proof of Step 1

 1 t

= ()k1n ; k1(0; 0) = 1 :

Noti e that, although   1 and  0 in the blow-up argument des ribed in x5.2, the same argument applies without any hange to the present situation. The urvature bound near the paraboli boundary guarantees that 1 is a omplete, unbounded urve. Furthermore, its total urvature is less than or equal to  . Now, we an represent 1 (
; t) as the graph of a smooth fun tion

© 2001 by Chapman & Hall/CRC

6.3.

155

Shrinking to a point

v in R  (

1; 0℄ whose derivatives satisfy   jvxj 6 tan 1  2  ; vx(0; 0) = 0 ; vxx(0; 0) = 1 ;

for all t, and it satis es the equation

vxx 1 + vx2 ; where  = tan 1 vx + 0 , 0 2 [0; 2). The onstant 0 results from the hange of oordinates whi h puts 1 into a graph over the x-axis. Now, as before, we observe that the non-negative fun tion w = vx vt = ()

satis es the uniformly paraboli equation 

wt = ()

wx  1 + w2 x :

By Moser's Harna k inequality ([88℄), we on lude that w is a onstant. The ontradi tion holds. First of all, we observe that it suÆ es to show Step 2 holds for t2 = ! and t1 , a xed time very lose to !. Lemma 6.5 holds trivially if t2 < !. Se ond, in view of Theorem 6.4, it suÆ es to show Step 2 for disks of the form D" (Q) where Q 2 fQ1 ;


; Qm g and " is suÆ iently small. In the following, we shall establish the validity of Step 2 on D" (Q)  [t; !) where " and t will be hosen suÆ iently lose to 0 and !, respe tively. Let Q = Qj be xed and "0 is so small that D"0 (Q) ontains no singularities other than Q, and (
; t) \ D"0 (Q) is onne ted for all t 2 [t; !). Let  =  \ D"0 (Q) and j =
 \ D" (Q) n D" +1 (Q) , where "j = (100) j "0 . Sin e (
; t) tends to  smoothly away from Q and the total absolute urvature of

(
; t) is uniformly bounded, there exists ` su h that Proof of Step 2

j

j

Z

 j

© 2001 by Chapman & Hall/CRC

 ; for all j > ` : jkjds < 200

156

CH. 6.

A Class of Non- onvex Anisotropi Flows

We hoose t su h that Z

` (t)

 ; for all t 2 [t; !) : jkjds < 100

(We shall restri t t further as we pro eed.) From now on, we write " = "` . Observing that the total absolute urvature along ` (t) is very small and D" +1 (Q) is very small ompared to D"1 (Q), the two ar s omprising `(t) are virtually straight rays emitting from the origin, and are almost perpendi ular to the boundary of D"1 (Q). We would like to determine a time-varying foliation F (
; t) whi h meets the following requirement: for all t 2 [t; !), 1

`

(i) the endpoints of 0 (
; t)  (
; t) \ D"1 (Q), P (t) and P (t), lie outside F (
; t), (ii) F (
; t) overs a xed small neighborhood of Q, (iii) 0 (
; t) passes through the leaves of F (
; t) without tangen y, and,
(iv) for ea h ,  t; F (
; t) \ D"1 (Q) 6 n 4 . 1

2

Let  = =100. We rotate the oordinate axes so that the tangent angle (
; t) of 0(
; t) satis es 



lim inf (P; t) : P 2 0 (
; t) = 0 : t"!

(6.6)

Then (iii) and (iv) will be satis ed if F is de ned in su h a way that its tangent angle  satis es 2; (n + 1) 4  2 +  and, whenever F meets (
; t), 2



 2

© 2001 by Chapman & Hall/CRC

h



i

 + 2 2 2 : ;



(6.7) (6.8)

6.3.

157

Shrinking to a point

In our onstru tion, ea h leaf of F (
; t) will be an embedded losed

urve. Then, (i) and (ii) will also be satis ed if the boundary leaves F1 (
; t) and F2 (
; t) separate P1 (t) from Q and P2 (t) from Q, respe tively. Let Pi (i = 1; 2) be the endpoints of  \ D"1 (Q). A

ording to our hoi e of "1 , for a number Æ whi h is very small ompared to "1 , the ir le C2Æ (Pi ) interse ts 0 (
; t) at exa tly one point Xi , i = 1; 2. Let `i = Ai Bi be the hord of D"1 (Q) formed by the straight line passing through Xi and pointing at the dire tion ( os i ; sin i ), where i = maxf(Xi ; t) =2,  g; i = 1; 2. We assume that ( os i ; sin i ) points outward at Bi and inward at Ai . De ne a ve tor eld v0 = ( os ; sin ) along 0 (
; t) by setting

n

 = max 

 ; 2

o

 ;

(6.9)

and  = i , along `i for i = 1; 2. Noti e that v0 is transversal along 0 (
; t) and it satis es (6.7) and (6.8). The ve tor eld is Lips hitz

ontinuous. We repla e it by a smooth approximation and still denote it by v0 . By (6.6) and (6.9), one an he k that the two hords `1 and `2 are disjoint. (In fa t, the hoi e of D"1 (Q) and D"`+1 (Q) shows that 0 (
; t) is almost perpendi ular to the boundary of D"1 (Q). It is shown in [92℄ that the angle between 0 (
; t) and D"1 (Q) at Pi (t), i = 1; 2; is bounded between =2  =50. When  =  =2, the hord `1 or `2 is very short. On the other hand, if  =  at both endpoints, large portions of the hords `1 and `2 lie inside dierent omponents of D"1 (Q) n 0 (
; t).) Also, together with the _ _ ar s A1 A2 and B1 B2 , they bound a sub-region of D"1 (Q), R, su h that D"1 =100 (Q) is ontained in R. Sin e v0 points inward at A1 and A2 , and outward at B1 and B2 , we an de ne a smooth v0 along _ _ the ar A1 A2 (resp. B1 B2 ) so that it always points inward (resp. outward), and (6.7) ontinues to hold.

© 2001 by Chapman & Hall/CRC

158

CH. 6.

A Class of Non- onvex Anisotropi Flows

Figure 6.1 The large disk in this gure is

D"1 (Q) and the small one is D"1 =100 (Q).

By the Tietze Extension Theorem, we an extend v0 to a ve tor eld (still denoted by) v0 on R satisfying (6.7). Sin e v0 is smooth, we may assume without loss of generality that the extended v0 is also smooth. Any integral urve of v0 is transversal to (
; t), and _ _ its endpoints lie on the ar s A1 A2 and B1 B2 . We may extend ea h integral urve outside R by smoothly atta hing a line segment to ea h endpoint. Let `i = Ai Bi be the extended line segment. With _ _ a ne essary restri tion on Æ, we may assume A1 A2 and B1 B2 lie on C"1 +2Æ (Q). In this way, we have found a foliation of extended _ integral urves on the region R bounded between `1 ; `2 ; A1 A2 , and _ B1 B2 . Finally, we extend ea h integral urve outside D"1 +2Æ (Q) by

losing it up with a large ar to obtain a foliation F whi h onsists 0

0

0

0

0

0

0

0

© 2001 by Chapman & Hall/CRC

0

0

0

0

0

0

0

6.3.

159

Shrinking to a point

of embedded losed urves with `01  F0 and `02  F1 . F0 separates D2Æ (P1 ) from D2Æ (Q) and D2Æ (P2 ), and F1 separates D2Æ (P2 ) from D2Æ (Q) and D2Æ (P1 ). Here, Æ0 = CÆ and C < 1 is a onstant determined by . Let F(
; t) be the foliation obtained by solving (6.3) using ea h leaf F as the initial urve at t. We laim that there is an a priori uniform bound on the urvature of the leaves of this foliation outside D"1 +3Æ=2 (Q) in [t; t + ℄ for some  > 0. For, rst of all, there is a uniform bound on the urvature of the leaves outside D"1 +3Æ=2 (Q) at time t. Therefore, for any point X on F (
; t), one an nd a small square S` entered at X , whose side ` is less than Æ=2, su h that F (
; t) \ S` is the graph of a fun tion over one side of S`. When ` is suÆ iently small, the tangent angles of this graph are nearly

onstant. Using Proposition 1.10, the existen e of stationary ar s of (6.3) transversal to the graph, the Sturm os illation theorem, and paraboli regularity, we know that there is some  > 0 su h that the

urvature of F(
; t) \ S`=2 is uniformly bounded for all t 2 [t; t + ℄. Sin e X ould be any point on F (
; t)n D"1 +3Æ=2 (Q), the urvature of F (
; t) outside D"1+3Æ=2 (Q) is uniformly bounded in [t; !℄, provided we hoose t su h that t +  > !. Next, onsider the urvature of F (
; t) inside D"1+2Æ (Q). At t = t, 0

0

0

0

0

0

  t; F (
; t) \ D"1 +2Æ (Q) 6 n 4 4 + 3  ; by onstru tion. Sin e onvexn on ave ar s are nesting and the urvature of F (
; t) near the ends is uniformly bounded,


 t; F (
; t) \ D"1 +2Æ (Q) 6 n 4  for all t 2 [t; !℄ when t is lose to !. By indu tion hypothesis, the

urvature is also bounded in D"1 +2Æ (Q). By Proposition 1.2, the foliation F (
; t) exists in [t; !℄.


© 2001 by Chapman & Hall/CRC

160

CH. 6.

A Class of Non- onvex Anisotropi Flows

We still have to make sure that the endpoints of 0 (
; t) never tou h the foliation F (
; t). Consider the ir les C2Æ (P1 ); C2Æ (P2 ), and C2Æ (Q). Using them as initial urves starting at t, we solve (6.3) to obtain three ows. When Æ0 is suÆ iently small, the ows are shrinking and we an nd t lose to ! su h that they still ontain the disks DÆ (P1 ); DÆ (P2 ), and DÆ (Q), respe tively for t 2 [t; t+!℄. Sin e Pi (t) tends to Pi ; i = 1; 2, as t " !. We may assume that the endpoints of 0 (
; t) are ontained inside DÆ (P1 ) and DÆ (P2 ) during the same time interval. So, nally we have onstru ted a foliation and found a t su h that our requirements (i)|(iv) are satis ed. It is straightforward to bend the foliation to the left and right to obtain two foliations satisfying the assumptions in Proposition 6.3. By this proposition, the urvature of 0 (
; t) is uniformly bounded in DÆ (Q) for all t in [t; !). Hen e, Step 2 holds. 0

0

0

0

0

0

0

0

0

6.4

A whisker lemma

So far, we have shown that the ow (
; t) shrinks to a point as t " !. Further, sin e the number of in e tion points does not in rease, we may assume that it is always equal to N in [0; !). We shall ontinue to assume (
; t) is non onvex and, hen e, N > 2. We an de ompose

(
; t) into a union of onvexn on ave evolving ar s 1 (t);


; N (t) whose total absolute urvature is either less than  Æ or greater than  for some Æ > 0.

Lemma 6.7 The urvature on an ar the total absolute urvature of

© 2001 by Chapman & Hall/CRC

j (t)

j (t)

is uniformly bounded if

is less than



Æ.

6.4.

161

A whisker lemma

Let (t) be su h an ar . We an use the tangent angle  to parametrize it. The equation for the urvature, k(; t), is given by

Proof:

h

i

kt = k2 (k + ) + (k + ) :

(6.10)

Without loss of generality, let's assume the tangent angles along

(0) lie in [Æ=2;  Æ=2℄. Then, the tangent angles of (t) lie in the same interval for all subsequent time t. Equation (6.10) admits two stationary solutions k =




1 h  j jmax + maxjkjmax (0) sin  i :  sin Æ=2

At t = 0, k () 6 k(; 0) 6 k+ (), for all  in [Æ=2;  Æ=2℄. As k+ is non-negative and k is non-positive in [Æ=2;  Æ=2℄  [0; 1), by the omparison prin iple, we have h maxkmax (0) + j j i : jk(; t)j 6 min1 j jmax +sinÆ= max 2



Let be a losed urve. We all an ar  of a ni e ar (with respe t to a oordinate system) if (i) the urvature on is negative and bounded away from zero, and (ii) the tangents at the endpoints of are horizontal. Sin e the tangent angles (or normal angles) of ea h j (t) are stri tly nesting in time, for any given ni e ar (t0 ) of (
; t0 ), there exists a family of ontinuously hanging ni e ar s (t), of (
; t), t 6 t0 ,

onne ting (t0 ) to some ni e ar (0). Let (t) be su h a family of ni e ar s of (
; t); t 2 [0; t0 ). We de ne the Æ -whisker of (t) to be the set fX = (t)+ e :  2 (0; Æ) and e is the unit horizontal tangent pointing to the interior of (
; t)g.

© 2001 by Chapman & Hall/CRC

162

CH. 6.

A Class of Non- onvex Anisotropi Flows

Æ > 0 depending on (
; 0) su h that, for

(
; t0 ); t0 2 [0; !), its Æ-whisker is ontained

Lemma 6.8 There exists

(t0 ) on in the interior of (
; t0 ).

any ni e ar

Let (t) = (
; t) n (t) be the omplement of (t). We rst

onne t (t0 ) through ni e ar s to (0). Then, we de ne

Proof:

n

o

d(t) = min 1 (t) 1 (t) ;


where (t) = 1 (t), 2 (t) is a point on (t) and (t) = 1 (t),
2 (t) is a point on (t) satisfying 2 (t) = 2 (t). We shall prove the lemma by showing that d(t) is in reasing in [0; t0 ℄. To this end, let's take e = (1; 0). (The other ase e = ( 1; 0) an be handled similarly.) Let e(t) be the portion of (t) that is bounded between the two horizontal tangents lines passing the endpoints of (t). Every horizontal line segment with endpoints on (t) and e(t) lies entirely inside (
; t). Then n

o

d(t) = min e1 (t) 1 (t) ;


where (t) = (1 (t); 2 (t) is a point on (t) and e(t) = e1 (t),
e2 (t) is a point on e(t) satisfying e2 (t) = 2 (t). Geometri ally speaking, d(t) is the minimal distan e overed when we translate (t) to its right horizontally until it rst hits e(t). To show that d(t) is in reasing in time, it suÆ es to show that (t ) + d(t)e separates from e(t ) for t > t. Let P be a point on the interse tion of e(t) and (t) + d(t)e. We laim that, in a neighborhood of P , e1 (t ) and (t1 ) + d(t)e are disjoint for t > t. For, rst let's assume P belongs to, interior of (t) + d(t)e. Then we an represent (t ) and e(t ) lo ally at P on the graphs of two fun tions x = x1 (y; t ) and x = x2 (y; t ), respe tively. By (6.2), both x1 and 0

0

0

0

0

© 2001 by Chapman & Hall/CRC

0

0

0

0

6.4.

163

A whisker lemma

x2 satisfy the equation

"

#

x   x 2  12  2 x=y2 ( ) = 1 + t y 1 + (x=y)2 + ()

in [p2 "; p2 + "℄  [t1 ; t + "℄ for some small " > 0. Moreover, x2 > x1 + d(t) at t and x2 (p2  "; t ) > x1 (p2  "; t ) for t 2 [t; t + "℄. By the strong maximum prin iple, 0

0

0

x2 (y; t) > x1 (y; t) + d(t)

for (y; t ) in (p2 "; p2 + ")  (t; t + "). So, (t ) + d(t)e is separated from e(t ) in this neighborhood. In ase P happens to be one of the endpoints of (t) + d(t)e, we

an represent the two ar s of (
; t) in a neighborhood of P , where one onne ts (t ) and the other onne ts e(t ), as graphs of two fun tions y = y1 (x; t ) and y = y2 (x; t ), respe tively. By the same reasoning as before, (t1 )+d(t)e and e(t ) separate within this neighborhood instantly. Now, suppose that (t ) + d(t)e and e(t ) do not separate instantly. We an nd ftj g , tj # t, and fPj g on (t1 )+ d(t)e and e(t1 ) su h that fPj g onverges to some P0 on (t) + d(t)e. However, it means that (tj ) + d(t)e is not separated from e(tj ) in any neighborhood of P0 for all large j . This is ontradi tory to what we have proved. Thus, (t ) + d(t)e must separate from e(t ) for all t1 > t. 0

0

0

0

0

0

0

0

0

0



0

0

As an appli ation of the whisker lemma, we have:

Proposition 6.9 The total absolute urvature on a negative ar

t " !. Consequently,

(
; t) tend to 2 as t " !. tends to zero as

j (t)

the total absolute urvature of

Were there a negative ar whose total absolute urvature is greater than  in [0; !), we an nd a family of ni e ar s (t) on

Proof:

© 2001 by Chapman & Hall/CRC

164

CH. 6.

A Class of Non- onvex Anisotropi Flows

this ar in a suitably xed oordinate system. But the existen e of a Æ-whisker prevents the urve from shrinking to a point. Hen e, the total urvature along a negative ar must be less than  Æ. By Lemma 6.7, its urvature is uniformly bounded. Sin e the total length of (
; t) tends to zero, we have Z jkjds 6 onst.  L(t) ! 0

j

as t " !. 6.5



The onvexity theorem

Finally, we nish the proof of Theorem 6.1 in this se tion. Assume that the ow resists onvexity until its very end. By a proper hoi e of oordinates we may assume, in view of Proposition 6.9, that, on every (
; t), there is an ar of negative urvature with normal image always ontaining the north pole and shrinking to it as t " !. Let e be an area preserving expansion of . We shall onsider two ases separately.

Case I

There exists ftj g; tj

e(
; tj ) is uniformly bounded.

Case II

" !,

su h that the diameter of ea h

The diameter of e

(
; t) tends to in nity as t " ! .

As the total absolute urvature of (
; t) approa hes 2 at the end, we may assume that (
; t) is the union of the graphs of two fun tions: u1 (x; t) 6 u2 (x; t) for x 2 [a(t); b(t)℄. All ar s of negative urvature will eventually lie parallel to the x-axis, for otherwise, there would be an ar of positive urvature whose total urvature is stri tly less than , and yet has a positive lower bound. However,

© 2001 by Chapman & Hall/CRC

6.5.

165

The onvexity theorem

this is impossible by Lemma 6.7 and Theorem 6.6. We shall rst show that Case I is impossible. For ea h (
; tj ), we an x a oordinate system su h that the origin is lo ated on an ar of negative urvature and its unit tangent is e1 = (1; 0), and the normal n = (0; 1) points to the interior of

e(
; tj ). In Case I, there exists M > 0 su h that e(
; tj ) is bounded between x = M and, ue1 (x; tj ) > "j with "j ! 0 as j ! 1. Here ue1 orresponds to u1 . For ea h tj , we hange the time s ale of e by setting

(
;  ) =

s

A(0)

(
; t) ; A(tj )

where A(t) is the area en losed by (
; t) and  = A(0)(t Then, satis es 8 > <  > :

t



=  k + 

s

for  in 0; A(0)(! By (1.19) , A(tj ) =

tj )=A(tj )

 Z 2 0




A(tj ) = lim t "! ! t j



Z 2 0

A(tj ) ; A(0)

(6.11)

.

()d

Therefore, j

=

n;

(
; 0) = e

(t j )



A(! )



tj )=A(tj ).

!

tj




Z !Z tj

dsdt :

()d :

For all suÆ iently large j , we may assume that (6.11) is valid in [0; a℄ where a = A(0)

© 2001 by Chapman & Hall/CRC

Z

2 ()d


1

:

166

CH. 6.

A Class of Non- onvex Anisotropi Flows

Let h be the solution for the mixed problem  h = 14 min x 2

8 h > > > > t > > >
> > > > > > : h

where f is given by 8 1 (x M 2) + 1 ; > > > > 2 > >

M

( ) = >0 > > > > > > > :

1 (x + M + 2) + 1 ; 2

For ea h Æ 2 solution of (6.12) satis es

Lemma 6.10

(i) (ii)

M

(6.12)

2 [M; M + 2℄ ;

x

2(

M; M ;

x

2[

M

(0; 1),

+ 2)  [0; 1) ;

2; M + 2℄ ;

x

> >


0 su h that the

) >  for all (x;  ) 2 [ M 2; M + 2℄  [Æ; 1), and 1; M + 1℄  [0; 1). x (x;  ) 6 1 for all (x;  ) 2 [ M

(

h x;  h

(i) is a dire t onsequen e of the strong maximum prin iple. To prove (ii), we rst observe that h(
; t) is onvex for all t. Therefore, for all x 2 [ M 1; M + 1℄,

Proof:

h

(

x x; 

x;  ) ) 6 distf1x; h((M + 2)g

6 1:

© 2001 by Chapman & Hall/CRC



6.5.

167

The onvexity theorem

Fix Æ = minf1; ag and " < =2. We hoose a smooth approximation of f; g, satisfying g > 0, f 6 g 6 f + ", and g((M + 2)) = 1. Denote the solution of the mixed problem of (6.12) where f is repla ed by g by eh. Then, eh satis es 00

e h

(x;  ) >  for all (x;  ) 2 [

M

2; M + 2℄  [Æ; 1)

and e h

(

x x; 

) 6 1 for all (x;  ) 2 [

M

1; M + 1℄  [0; 1)

for the same Æ and . Moreover, by applying the strong maximum prin iple to the se ond derivatives of eh, we know that (x;  ) > k0 > 0 ; for all (x;  ) 2 [

e k

M

1; M + 1℄  (0; 1) ;

for some k0 , where ek(
;  ) is the urvature of the urve (x; eh(x;  )). Setting w = eh 2", by (6.12) we have w 

2 = 14 min xw2 6 21 min 1 w+xxw2 x

:

By the onstru tion of eh, we an nd 0 > 0 su h that w 

6  1 w+xxw2 +  (1 + wx2 ) 12 x

1; M + 1℄  [0; 1) for all  2 [0; 0 ℄. For large j , e (
; t) is bounded between x = M and lies above the graph of g 2". Let's onsider the ow (
;  ) and look at the ar ontaining the origin. It is the graph of some fun tion u(x;  ). in [

M

© 2001 by Chapman & Hall/CRC

168

CH. 6.

A Class of Non- onvex Anisotropi Flows

Sin e the urvature is negative there, by (6.3) and (1.3) we have u 



6  1 + 6

u 2 1=2 x

C max



as the urvature is always bounded on this ar . Therefore, at time  , this subar lies below the line y = C j jmax . On the other hand, by applying the omparison prin iple to u and w, we know that this ar must lie above the graph of w at any  . Taking, in parti ular,  = Æ , we have ( ) > eh(x; Æ) 2" >  2" :

u x; Æ

Hen e, 

2" 6  j jmaxÆ :

However, the right-hand side of this inequality tends to zero as j ! 1, and the ontradi tion holds. So Case I is ex luded. Next, we onsider Case II. The dieren e U = u2 u1 satis es U t

2

= A(x; t) xU2 + B (x; t) U x

(6.13)

;

for some A(x; t) > 0 and B (x; t), x 2 [a(t); b(t)℄. Lemma 6.11 There exists one lo al maximum in

t0

lose to

[a(t); b(t)℄

!

su h that

(
)

U ;t

has only

.

We shall prove the lemma by showing that U has no lo al minima in (a(t); b(t)) for all t > t0 where t0 is lose to !. By applying Proof:

© 2001 by Chapman & Hall/CRC

6.5.

169

The onvexity theorem

the Sturm os illation theorem to the equation satis ed by U=x, we know that the number of lo al minima is nite and nonin reasing in t. Moreover, through every minimum P there passes a uniquely determined dierentiable urve X (t) onsisting of lo al minima of

(
; t). Were the lemma not true, there exists su h path X (t) for all t 2 [t0 ; ! ). By (6.13),

U
dX U dU (t) > 0 : X (t); t = X (t); t + X (t); t dt t x dt

Hen e, 0 < U X (t0 ); t0




6U




X (t); t

!0

as t " !, whi h is impossible. Lemma 6.12 Let

be any embedded losed urve whi h is the union

of the graphs of two fun tions that

 u2

U

u1 q2

where

q1

and

 u1 and u2 , u1

6 u2

, over

[a; b℄.

Suppose

has only one lo al maximum. Then,

q1

6 23 (b

q2 , q1 < q2 ,

verti al lines whi h separate

a) ; are respe tively the

x- oordinates

of the

into three pie es, the left and the right

ones having one quarter of the area.

Let q be the x- oordinate of the verti al line whi h divides , or, more pre isely, the region en losed by , into two pie es of equal area. The verti al lines through q1 ; q; and q2 divide the region into four parts of equal area. Call them P1 ; P2 ; P3 , and P4 , and denote their widths, that is, the dieren es between the q s and a; b, by  w1 ; w2 ; w3 , and w4 . The maximum thi kness, Umax = max u2 (x) u1 (x) : x 2 [a; b℄ , of is realized either in P1 ; P4 or P2 ; P3 . Suppose that Umax is realized in P1 . By Lemma 6.11, U de reases

Proof:

0

© 2001 by Chapman & Hall/CRC

170

CH. 6.

as we move through the other w3

6

w4 .

q1

=

w2

6

2

6 Suppose that = 1;

P s.

As all

Pi s

have equal area,

w2

6

Therefore, q2

Pi , i

A Class of Non- onvex Anisotropi Flows






;

3 2 3

+ w3

(w2 + w3 + w4 ) (b

a) :

is realized in

Umax

P2 .

4, is equal to 1. Divide

P2

We may assume the area of further into two parts by a

verti al line through the maximal thi kness. Denote their areas and widths by values of P2 ,

A1 , A3

U (x),

and

where

w21 ; w23 , x

respe tively. Let

U1

and

U

6

U1

on

w1 ;

U

>

U3

on

w23 ;

U

>

U1

on

w21 ;

6

U3

on

w3 :

w1

>

1

; w21

6

A1

; w3

>

1

Therefore,

U1

U1

and w23

6

A3 U3

U3

It follows that w21

6

A1 w1

6

w1 ;

w23

6

A3 w3

6

w3 :

and

© 2001 by Chapman & Hall/CRC

be the

denotes the left and the right endpoints of

respe tively. By Lemma 6.11,

U

U3

:

6.5.

171

The onvexity theorem

As before, w3 6 w4 , and so,

q2 q1 = w21 + w23 + w3  w1 + w2 + w3 < 2w1 + 2w4 ; whi h implies that 3(q2

q1 ) < 2(w1 + w2 + w3 + w4 ) :



In Case II, the diameter of e(
; t) tends to in nity as t " !. For ea h e(
; t), there exists a translation so that the origin is lo ated on an ar of negative urvature with tangent e1 . When there exist M; , and ftj g; tj " !, su h that ea h e(
; tj ), after a translation, lies between the lines y = (x M ), y = (x + M ), and ue1 > "j , with "j ! 0 as j ! 1, we an follow the argument in Case I to draw a ontradi tion. Here, ue1 orresponds to the dilation of u1 . Therefore, we may assume, for all large M and small > 0, ea h e(
; t), after a suitable translation, tou hes one of the lines y = (x M ). Let b(
; t) be the length-preserving expansion of (
; t). By noting the fa ts that e(
; t) has xed en losed area and its total absolute

urvature tends to 2, we know that b(
; t) must ollapse into a line segment on the x-axis after a suitable translation depending on t. Let `1 and `2 be the verti al lines x = a(t) and x = b(t), respe tively. Any verti al line x = ` between `1 and `2 uts u1 (resp. u2 ) at one point transversely. Denote its slope by tan i , ji j < =2, i = 1; 2. We laim that, for ea h xed " > 0, there exists t su h that, if ji j > " for some i, the minimum distan e between ` and `1 , and the distan e between ` and `2 , is less than "(b(t) a(t)) for all t in [t ; !). For, we an look at the length-preserving expansion b(
; t) whi h ollapses into a line segment after a time-dependent translation. Sin e its total absolute urvature tends to 2, the slopes at the 00

00

© 2001 by Chapman & Hall/CRC

172

CH. 6.

A Class of Non- onvex Anisotropi Flows

interse tions of any verti al lines based at a point [ba(t) + "; bb(t) "℄ and b(
; t) must tend to zero as t " !. Hen e, there exists t su h that ji j 6 j tan i j < ". Now, for some small " to be spe i ed below, we onsider t > maxft0 ; t g, where t0 is hosen a

ording to Lemma 6.11. Let  1 max  + " A(t) bt = t + 2 Z min : ()d 00

00

We have A(bt)

Z bt Z 

= A(t)

t

Z

= A(t) Z bt Z t





k + dsd

  ()d 21 + max " Z A(t) min ()d

dsdt

h > 12 max " min

max " Zj jmax max L( )iA(t) ; + [t;b t℄ 2 min d

1

where L(t) is the length of (
; t). As L(t) tends to zero as t " !, for small " and t lose to !, 5 A(bt) > A(t) : (6.14) 11 Next, for a xed t, we onsider q1 and q2 for (
; t) as determined in Lemma 6.12. The verti al lines passing q1 and q2 divide (
; t),  2 [t; bt℄, into, at most, three parts. At the left pie e, either (i) for all  2 [t; bt℄, the angles  made with the tangents at the interse tion of x = q1 and (
;  ) do not ex eed ", or

© 2001 by Chapman & Hall/CRC

6.5.

173

The onvexity theorem

(ii) there exists  2 [t; bt℄ su h that one of these angles satis es jj > ". In the rst ase, denote the area of this pie e by A` ( ). We have dA` ( ) = d

>

Z 2+  

Z 2 " +"

()d

Z C (
; )

ds

()d j jmax
L( ) ;

where C (
;  ) is the ar (
;  ) \ f(x; y) : x 6 q1 g. We have A` (bt)

6 A` (t)

 Z 2 " +"

  d j jmax max L( ) 12 + max " ZA(t) min d

# 1 R d  Z 2 " d j j max L( ) 1 + max " max 2 min +" d 4

=

ZA(t)

6

ZA(t)

d

"

"

(1 + 2" )max" + 12 j jmax max L( )+ #

 "j jmax max max L( ) : min Noti e that we have used (6.2) to get Z 2 

Z  = 21



2

0

:

After a further restri tion on t, we nd that, for small ", A` (bt) < 0 ;

© 2001 by Chapman & Hall/CRC

174

CH. 6.

A Class of Non- onvex Anisotropi Flows

whi h is impossible. So, with this hoi e of ", Case (ii) must hold. In other words, there is some  2 [t; bt℄ su h that a ( )

q1

6 

" b( )

"w( ) :




a( )

Clearly, under (6.3), straight lines translate in onstant speed. By

omparing (
; t) with the evolution of the verti al line at b(t), we have, for t1 < t2 , a(t2 ) > a(t1 )

j jmax(t2

t1 ) ;

b(t2 ) 6 b(t1 ) + j jmax (t2

t1 ) :

Therefore, q1

a(b t)

6 6 6 6

q1

a( ) + j jmax (b t

"w( ) + j jmax (b t )

"w(t) + 2"j jmax (

) t) + j jmax (b t

"w(t) + (1 + 2")j jmax (b t ) :

)

Similarly, we have b(b t)

q2

6 "w(t) + (1 + 2")j jmax (bt

) :

It follows from the isoperimetri inequality Cw2(t) > A(t) that we

© 2001 by Chapman & Hall/CRC

6.5.

175

The onvexity theorem

have, by Lemma 6.12, w(b t)

6

q2

= q2

6

j j

q1 + 2"w(t) + 2(1 + 2") max (b t

j j

q1 + 2"w(t) + 2(1 + 2") max

2

)

max "  ZA(t) + 2 min d

1





e 3 + 2" + Cw(t) w(t) ;

where Ce is a positive onstant depending only on  and . As w(t) tends to zero, for every suÆ iently small ", there exists t suÆ iently

lose to ! su h that 2 + 2" + e w(t) 6 2  100 : 3 3 99 Therefore, 200 w(t) : w(b t) 6 297 Taking t1 = t , 1 max " ZA(t) ; for j > 1 ; + tj +1 = tj + 2 min d we have A(tj +1 ) w2 (tj +1 )

> >

So,

A(tj +1 ) w2 (tj +1 )

5 11
A(tj +1 ) 200 2 w2 (tj +1) 297

1:002 wA2((ttj )) j

> (1:002)j wA2((tt)) ! 1 as

© 2001 by Chapman & Hall/CRC



:

j

!1;

176

CH. 6.

A Class of Non- onvex Anisotropi Flows

and yet, on the other hand, A(tj +1 ) w2 (tj +1 )

!0;

as j

!1;

as b(
; t) onverges to a line segment. This ontradi tion nally shows that Theorem 6.1 must hold.

Notes The ontent of this hapter is based on Grayson [66℄, Angenent [13℄, [14℄, Oaks [92℄, and Chou-Zhu [36℄. In parti ular, Theorem 6.4 is taken from [13℄, Theorem 6.6 from [92℄, and Theorem 6.1 from [36℄.

The limit urve

 . Theorem 6.4 ontinues to hold for immersed

losed ows of (1.21) on a surfa e under assumptions (i){(vi) (see Notes in Chapter 1). The behaviour of this ow near the singularities is studied in some depth in [14℄ and [92℄. Let's follow the latter and des ribe a basi result. First, we all [a; b℄ a singular interval of the

ow if it is the largest interval satisfying the following two onditions: (a) k([a; b℄; t) be omes unbounded as t " ! and (b)  ([a; b℄) is a singularity. Let Q =  (p0 ). For any " > 0, we denote by [a" ; b" ℄ the maximal ar ontaining p0 over whi h onverges entirely inside N" (Q). We have Theorem Let (
; t) be a maximal ow of (1.21) where (i){(vi) hold and ! is nite. Then, (A) if (
; t) is embedded, then it shrinks to a point as t " ! , (B) let Q =  (p0 ) be a singularity. Then, p0 is ontained in some singular interval I and either (a) there is a self-interse tion in

([a" ; b" ℄; t) onverging to Q (\a loop ontra ts"), or (b) there are p1 ; p2 62 I su h that  ([p1 ; p0 ℄) =  ([p0 ; p2 ℄) (\parametrization doubling"). It is ommonly believed that sub ase (B)(b) annot happen.

© 2001 by Chapman & Hall/CRC

6.5.

The onvexity theorem

177

Theorem 6.1 was rst onje tured in Gage [57℄ when  0. After proving that the ow shrinks to a point, the proof of onvexity follows Grayson [66℄ losely. Nevertheless, one

an show that Hamilton's approa h (see Chapter 5) an be extended to the anisotropi ase. In fa t, Zhu [113℄ has used this approa h to establish the onvexity and asymptoti behaviour of the ow (6.3) on a surfa e. Convexity result.

© 2001 by Chapman & Hall/CRC

Chapter 7

Embedded Closed Geodesi s on Surfa es As an appli ation to the theory of urve shortening ows developed in the previous hapters, we shall prove the existen e of embedded

losed geodesi s on a losed surfa e. A standard approa h for nding losed geodesi s is to look for

urves whi h minimize the length among a given homotopy lass. However, for a simply- onne ted surfa e, the homotopy lass is trivial and this approa h fails. When the surfa e is onvex, Poin are proposed to look for a geodesi by minimizing length among all embedded losed urves whi h divide the surfa e into two pie es ea h having total Gaussian urvature 2. Gage [56℄ found a ertain urve shortening ow whi h preserves urves in this lass. It turns out that this ow does exist for all time and sub onverges to an embedded

losed geodesi . For a general losed surfa e, the same proof shows that the standard urve shortening ow either exists for all time or shrinks to a point in nite time. Together with a topologi al minimax argument, one an show that there are always three embedded

losed geodesi s on a 2-sphere|a result rst formulated by Lusternik and S hnirelmann in 1929.

179 © 2001 by Chapman & Hall/CRC

180 7.1

CH. 7.

Embedded Closed Geodesi s on Surfa es

Basi results

Let M be an oriented smooth surfa e. Here, surfa e always means a two-dimensional Riemannian manifold with the metri g. Consider a family of smooth losed urves = (u; t) : S 1  [0; T ) ! M . Denote by  =t = (=t) its velo ity ve tor eld and  =u =

(=u) its tangent ve tor eld. The unit tangent ve tor is given by u =j u j and the unit normal ve tor is hosen su h that ( ; ) agrees with the orientation of the surfa e. Given a smooth fun tion F de ned in M R, onsider the Cau hy problem 



N

 = F ( ; k) t

N;

0

embedded losed ;

T

TN

(7.1)

where k is the geodesi urvature of the urve . In the following, we let s = j u j. Lemma 7.1

Let

(
; t)

be a solution of (7.1). Then, the following

hold: (i)

(ii)

(iii)

(iv)

(v)

s = t

T

F ks,

 = Fs t

N,

 i = kF , t s s

h

;

k = Fss + k2 F + KF , t d dt

Z

kds =

(
;t)

© 2001 by Chapman & Hall/CRC

R

KF ds,

7.1.

181

Basi results

(vi) where

d dt

Z

R

ds =

(
;t)

(
;t)

s = s(
; t)

F kds,

is the ar -length parametrization of

the Gaussian urvature of Proof:

(
; t)

and

K

is

M.

Re all that =s = s 1 =u. Let 

2 = tu

rZ

2 = ut

r u

Z = 









t


 ; u 
 ; t ;

where r is the Levi-Civita onne tion on M . From tu = ut and the Frenet formulas (1.1), whi h hold on surfa es, we have (i) and (ii). Next,   1 f  i f = t s t s u

h

1   f 

;

= kF

s u t

f ; s

by (i). Hen e, (iii) follows. In addition, by using the de nition of the Riemannian urvature tensor and (iii), we have  (k t

N)

© 2001 by Chapman & Hall/CRC

=

rZ rT T

=

rT rZ T + r Z;T T + R(Z; T )T [

℄

182

CH. 7.

=

rT

=

Fss

Fs

T

Embedded Closed Geodesi s on Surfa es

N
+ r[

=t;=s

Fs k

℄

T + R Z; T
T

T + F k2N + R Z; T
T :

On the other hand, by (7.1),

N
= k N + k tN : Taking the inner produ t with N yields (iv). Finally, (v) and (vi)  k t

t



follow from (i) and (iv).

Now we show that the solution of (7.1) exists for some positive time.

Let the initial urve

0

be smooth and losed.

its parameterization to an immersion



0



S1

:

[

We extend

!M

1; 1℄

with

= 0 . Then, any smooth losed urve whi h is C 2 - lose to

f0g

an be parametrized as

S1

f with f (u) < 1.

f (u)

Consider a smooth solution

0 .

starting at

image under




 u; f (u; t)



=

 (u; f (u))

(
; t)

:

for some

S 1  [0; T )

C 2 -fun tion

!M

of (7.1)

For small t, the solution an be represented as the of the graph of a fun tion

. The pull-ba k of the metri

g

f (u; t), on

M

i.e.,

under

(u; t) 

=

is given

by

  (g ) = A(u; v )(du)2 + 2B (u; v )dudv + C (u; v )(dv )2 ; where

A; B , and C

Setting

are smooth in

` = A + 2Bfu + Cfu2 ,

S1  [

1; 1℄, and

the tangent

D = AC B 2 > 0.

T and normal N to (
; t)

are given, respe tively, by



T

=

d `

N

=

d

n

1 2

`D

© 2001 by Chapman & Hall/CRC

 u


+ fu

1 2


 v

B + Cfu

and




u

+

A + Bfu


 o

v

;

7.1.

183

Basi results

where

A; B; C ,

and

D

are evaluated at (u; f (u; t)). By the Frenet

formulas, we have 3 1 2D2

k=` where

fuu + P

P; Q; R,

and

S

2 3 + Qfu + Rfu + Sfu




;

(7.2)

are smooth fun tions evaluated at (u; f (u; t)).

The verti al velo ity of

(
; t)

is given by

ft =v .

So, its normal

velo ity is



  g ft

1 1 2 D 2 ft

`

=

1h 2


 ; `D v

B + Cfu

f

u

+

A + Bfu


 i

v

:

It follows from these omputations that if




(
; t) solves (7.1) if and only

solves

ft where

=

1

`2 D

`; D;




1 2 F (; k )

:

et . are evaluated at

>From now on, we shall assume

where

(7.3)

f (u; t). F

in (7.1) is of the form

F

=

ak ;

a

is a smooth, positive fun tion in

(7.4)

M.

Then, equation (7.3)

is a quasilinear paraboli equation under this assumption. It follows

x

from Fa t 3 in 1.2 and the proof of Proposition 1.2 that the following result holds.

Consider the Cau hy problem for (7.1) where F satis es (7.4) and 0 is a smooth, losed urve. Then, it admits a unique maximal solution (
; t) in M  [0; !). Moreover, when ! is nite, the geodesi urvature of (
; t) be omes unbounded as t " !.

Proposition 7.2

© 2001 by Chapman & Hall/CRC

184

CH. 7.

Embedded Closed Geodesi s on Surfa es

The next ru ial lemma shows that the total absolute of the solution remains bounded on any nite time interval. Lemma 7.3

d dt

We have e

t

Z



(
;t)

jkjds 6

2e

t

X Fs (uj (t); t) ; j

where the summation in the right-hand side of this inequality is over all in e tion points uj (t) of (
; t) and  = (aK )max . Noti e that the Sturm os illation theorem an be applied to the equation in Lemma 7.1 (iv) to show that the number of in e tion points on (
; t) is nite for all t 2 (0; !). Proof:

d dt

By Lemma 7.1,

Z

(
;t)

jkjds

=

Z X 2 Fs (uj (t); t) + aKkds

6

Z X 2 Fs (uj (t); t) + (aK )max

k>0

(
;t)

Z

aKkds k60

jkjds ;

and the lemma follows. The following lemma is similar to Proposition 1.10.



There exist positive onstants Æ and C depending only on M and a su h that, for any solution (
; t) of (7.1), Lemma 7.4




(
; t)  NC pt (
; 0) ;

for 0 6 t < minfÆ; !g. Let P be any point in M whi h does not lie on the initial

urve (
; 0). Let d(t) be the distan e from P to (
; t). Sin e the

Proof:

© 2001 by Chapman & Hall/CRC

7.1.

185

Basi results

solution is smooth, d is Lips hitz ontinuous in t. Let 0 > 0 be the inje tivity radius of M. That means the exponential map expP : TP M ! M is an embedding on the disk D0

entered at the origin of TP M. If, at some instant t, the distan e fun tion is less than 0 =2, we an hoose Q on (
; t) whi h minimizes
dist (
; t); P . This point must lie in the image of D0 under the exponential map. Let (r; ') be the geodesi polar oordinates at P . We an represent the solution near Q as a graph r = f('; t), where Q =
d(t); '0 . Then f('0 ; t) = d(t) and f='('0 ; t) = 0. Moreover,  2 f='2 ('0 ; t) > 0. In these geodesi polar oordinates, the metri g is given by g = (dr)2 + A(r; ')(d')2 : By a dire t omputation, the geodesi urvature of the graph of r = f('; t) is 1

A2 k= 3 (A + p2 ) 2

2f '2

Ar  f 2 A '

A' f 2A '

1 A 2 r

!

;

where A' and Ar are evaluated at r = f('; t). Hen e, the geodesi

urvature of (
; t) at Q is at least the urvature of the geodesi ir le with radius d(t) entered at P . On the other hand, it is not hard to see that the urvature of a geodesi ir le with radius r, r 6 0 =2, is not less than 2 =(2r) for some depending only on the surfa e. Therefore, by (7.1), the distan e fun tion satis es C0 d (t) > ; (7.5) d(t) 0

whenever d(t) 6 0 =2. Here, C0 depends on M and F only. Now, let Æ = 20 =(8C0 ). If P lies on (
; t) for some t, then d(t) = 0. Integrating (7.5) gives 0 = d2 (t) > d2 (0)

© 2001 by Chapman & Hall/CRC

2C0 t ;

186

CH. 7.

Embedded Closed Geodesi s on Surfa es

p

whi h means that P lies in a 2C0 t neighborhood of (
; 0). 7.2



The limit urve

In this and the next se tion, we shall show that the ow (7.1) shrinks to a point when ! is nite. As usual, we shall assume this is not true and draw a ontradi tion by analyzing the ow near a singularity. First of all, when 0 is embedded and losed, we note that (
; t) is also embedded. This is a onsequen e of Proposition 1.5. Suppose ! is nite. By the same argument as in the proof of Chapter 6, as t " !, the solution onverges in the Hausdor metri to a limit

urve  whi h is the image of a Lips hitz ontinuous map from S 1 to M . Moreover, after a slight modi ation of the proof of Theorem 6.1, we dedu e from Lemma 7.3 that there exists a nite number of singularities fQ1 ;


; Qm g on  , su h that  n fQ1 ;


; Qm g

onsists of smooth urves. Away from the singularities, the urve

(
; t) onverges to  smoothly. Furthermore, for any P in  n fQ1 ;


; Qmg, there exists a neighborhood U of P and t0 2 [0; !) su h that U \ (
; t) is a onne ted evolving ar for all t > t0 . On the other hand, for any singularity Q, we an nd t1 lose to ! and a small 1 > 0 su h that the evolving ar (
; t) = D21 (Q) \ (
; t), where D21 (Q) is the geodesi disk of radius 21 at Q, is onne ted, and has exa tly two endpoints onverging to two distin t points on

 \ D21 (Q) as t " ! . We employ isothermal oordinates to express the evolution of (
; t) on M as an evolution in the plane. Without loss of generality, we may assume D21 (Q) is overed by isothermal oordinates. So, there is a onformal dieomorphism  from D21 (Q) to some open set V  R 2 su h that the metri g be omes g = J 2 (x; y )(dx2 + dy 2 ) ;

© 2001 by Chapman & Hall/CRC

7.2.

187

The limit urve

where J is bounded between two positive onstants. Let X = J 1 =x
and Y = J 1 =y be the unit ve tors. Let 0 (u; t) =  (u; t) . Sin e  is onformal,  (N ) = J 1 n0 , where n0 is the unit normal of the plane urve 0 (
; t). It implies that 0 (
; t) evolves a

ording to  0 1 = F n0 : (7.6) t

J

It is well-known that the Christoel symbols in isothermal oordinates are given by the following relations: J 

Y

rX X =

J

J 

rY X = JX

Y

Y

; ;

  rX Y = JJY X ;

rY Y =

where JX = rX J and JY = rY J . Let and X . So,



J 

X

J

(7.7)

X ;

be the angle between

T

T = os X + sin Y and

N=

sin X + os Y

(7.8)

:

By (7.7) and (7.8), we have k

N

= =

rT T 1

J

k0

JY J

 J

os  + X sin  N

J

;

where k0 is the urvature of 0 (
; t). Hen e, the Liouville formula, k

© 2001 by Chapman & Hall/CRC

=

1 0 k

J

JY J

J

os  + X sin  ; J

(7.9)

188

CH. 7.

Embedded Closed Geodesi s on Surfa es

holds. In view of this, the ow (7.6) an be written in the form

  0  0 = k + n0 : (7.10) t where  = (x; y) is pin hed between two positive onstants and

satis es (x; y;  + ) = (x; y; ) ; 8(x; y; ) 2 V

 S1 :

In other words, (7.10) is uniformly paraboli and symmetri . Without loss of generality, we may assume that (Q) is the origin and V ontains the unit disk entered at the origin D1 . Also, the interse tion of 0 (
; t) with D1=2 , A0 (t), and B 0 (t), onverges to two distin t points on  as t " !. 7.3

Shrinking to a point

Next, we'd like to derive an isoperimetri type estimate for the evolving ar 0 as des ribed in the last paragraph of the previous se tion. For ea h xed t 2 [t1 ; !), let be any embedded urve whose distin t endpoints lie on 0 (
; t) and whose interior is disjoint from 0 (
; t). Let L(t) be the lass of all su h urves. Letting L be the length of , we de ne  1 +1 G( ) = L2 A

and

n

o

g(t) = inf G( ) : 2 L(t) ;

where A is the area of the region en losed by and 0 (
; t). It is not hard to see that there exists a small "0 > 0 su h that, whenever g(t) < "0 , the in mum g(t) is taken on the sub lass

L0(t) =

n

2 L0 (t) : the endpoints of lie on

© 2001 by Chapman & Hall/CRC

0

o

(
; t) \ D1=4 :

7.3.

Shrinking to a point

189

If g(t) < "0 , the in mum g(t) is attained in L0 (t). It has onstant urvature and is perpendi ular to 0 (
; t) at its endpoints. Lemma 7.5

Let f j g be a minimizing sequen e of g(t) in L0 (t). Let Aj be the area en losed by j and 0 (
; t). We rst onsider the minimizing problem:

Proof:

n Lj = inf length of : is a urve in L0 (t) whose en losed area o with 0 (
; t) is equal to Aj :

It is well-known that this onstrained minimization problem is attained by a urve j0 whose endpoints interse t 0 (
; t) at right angles. Arguing as in x5.1, one an show that the interior of j0 does not tou h 0 (
; t). Hen e, j0 is a ir ular ar or a line segment whose interior is disjoint from 0 (
; t). Repla ing the minimizing sequen e f j g by f j0 g, we an argue as before that it sub onverges to a minimizer of g(t), whi h is again a ir ular ar or a line segment meeting 0 (
; t) at right angles and disjoint from 0 (
; t) at its interior points.



Now we ompute the rst and se ond variations at a minimizer 0 for a xed time. Here, the time variable is suppressed. For  2 [ 0 ; 0 ℄; 0 > 0. Let  be any one-parameter family of admissible

urves, A() and L() are, respe tively, the area en losed by  and 0 (
; ), and the length of  . For simpli ity, we assume the urvature of 0 ; k0 , is nonzero. The ase k0 = 0 an be treated similarly. Using polar oordinates at the enter of 0 , whi h is a ir ular ar ,  is given by the graph of r = r(; ) between  =  () and + = + ()  and 0 = (r; ) : r = jk0 j 1 ,  2 [ ; + ℄ . Sin e the fun tion r(; 0) is onstant and 0 is perpendi ular to

© 2001 by Chapman & Hall/CRC

190

CH. 7.

0 (
; t), we have

Embedded Closed Geodesi s on Surfa es

2r r =0; 2 =0  

and +  =0; =0;  

at  = 0. The urvatures of 0 (
; t) at  = + and  =  are given by k+0 and k0 , respe tively. They an be omputed as the graph of  = + () and r = r + (); 




or


 =  () and r = r  ();  ;

ex ept with a possible sign hange, sin e we keep the sign onvention for k0 in the ow (7.10). By a dire t omputation, we have  d2    r 2 r 2 = k+0 d +  + and  r 2  d2   ; = k0 r 2 d  at  = 0. Here and in the following subs ript, \ + " or \ " means evaluation at  = + or  . The omputation and result ould be easily expressed in terms of the velo ity v and the a

eleration z of  , r 2r v= and z = 2 ;   at  = 0. The length of  is given by L() =

© 2001 by Chapman & Hall/CRC

Z + r 

r2 +

 dr 2

d

d :

7.3.

191

Shrinking to a point

For simpli ity, we may assume the region en losed by  and 0 (
; t) is on the origin side of  . Then, A()

A(0) =

Z  Z  ( ) +

 ( )

0

r

r dd : 

By a straightforward omputation, we have,  = 0,

Lemma 7.6 At

dL = d

Z  +

d2 L = d2

Z  +

dA = d

1





jk j 0

1

d2 A = d2

jk j 0

we have

vd ; zd + jk0 j Z  +

 Z  +



Z   +



dv 2 d d



0 v2 + k0 v2 k+ +

vd ; zd +



;

and

Z  +



v2 d :

Sin e log G attains its minimum at 0 , we have 0 = =

d log G d =0 

2

L

+

1

1

 Z +

0



jk j(A + 1) jk jA 0

vd :

One an hoose the integral on the right non-zero. Therefore, 2

L

=

© 2001 by Chapman & Hall/CRC

1

jk jA(A + 1) : 0

(7.11)

192

CH. 7.

Embedded Closed Geodesi s on Surfa es

Next, by Lemma 7.6 and (7.11), d2 log G d2 =0

0 6

hZ = L2

2

L2

+



j j

zd + k0

 Z + 

vd

2

Z +   dv 2 

d 

d



0 v2 + k0 v2 k+ +

i

Z + Z +  1 1 v 2 d zd + + A + 1 jk j 0  

1  1 Z + vd2 (A + 1)2 jk0 j 

1 1

A

jk0 j

Z + 

zd +

Z + 

v 2 d



 1 Z + 2 1 vd + A2 jk j 0  Z = 2jLk0 j

+



h dv 2

d

i v 2 d

2 k 0 v 2 + k 0 v 2  L + +

Z 2 4 A + 1  + 1 vd : + 2jk j2 A2 (A + 1)2 0 

Choosing v

we have

q


=  jk0 j 1 os ; jk0 j 1 sin  ;  2 [  dv 2

d

  6 C1 jk1j2 + 1 0

; + ℄ ;

for some positive onstant C1 depending on . Noting that jk0 j = (+  )=L, we dedu e from (7.11) that i 2jk0 j Z +  dv 2 d 6 2C h1 + L2 : 1 L d A2 (A + 1)2  Thus,  L2 2 k0 (x ; y ) + k0 (x ; y ) 6 C 1 + (7.12) + + 2 L + A2 (A + 1)2

© 2001 by Chapman & Hall/CRC

7.3.

193

Shrinking to a point

for some onstant C2 depending on . Here, (x+ ; y+ ) and (x ; y ) are the position ve tors of the endpoints of 0 . Now we an state the isoperimetri type estimate. Lemma 7.7 There exists a positive onstant

Æ

su h that

0 on [t1 ; !).

g(t) > Æ >

Fix t 2 [t1 ; !) and let be the minimizer of g(t). For t lose to t, let t be the ontinuously hanging ir ular ar s or line segments whose endpoints lie on 0 (
; t) and t = . The time derivative of the length of t at t is the sum of the negative normal speed of 0 (
; t) at the endpoints of t . By noting the orientation of 0 (
; t), the length L(t) of t satis es Proof:

dL = dt t=t



(x ; y )k0 + (x ; y ;  ) + (x+ ; y+ )k+0 

+ (x+; y+ ; + + ) : A

ording to (1.19), dA = dt

Z



0 (
;t)



( 0 )k0 + ( 0 ; ) ds ;

where 0 (
; t) is the portion of 0 (
; t) whi h together with bounds the area A(t). Let (x; y) be any xed point on 0 (
; t). By a dire t omputation, d log G = dt t=t

2

L 

(x+ ; y+ )k+0 + (x ; y )k0

1 + (x ; y ;  ) + A(A + 1)

© 2001 by Chapman & Hall/CRC

Z

0 (
;t)




k0 + ds

(x+ ; y+ ; + )

194

CH. 7.

 > L2 (x

+ ; y+

1 + A(A + 1)

)k+0 + (x

Z

0 (
;t)

Embedded Closed Geodesi s on Surfa es

)k0

;y

;y ;





(7.13)

;


where L 0 (
; t) is the length of 0 (
; t). By Lemma 7.3, we know

0 (
;t)

6

(x; y )

(x; y)jk0 jds



max (x; y) (x; y) ;

0 (
;t) where C3 only depends on ; , and the initial urve. Without loss of generality, we assume  6 0 6 + . Then, Z

0 (
;t)

C3

(x; y)k0 ds =

Z 

+2

+ +

= (x; y)

(x; y)d 

+ + 

):

Thus, (7.13) an be written as d log G dt t=t

2h L

+

>

(x+ ; y+ )k+0 + (x

;y

 1 (x; y)  (+ A(A + 1)

)k0 

(x+ ; y+ ; + ) + (x

;y ;



)


 1  C3 max (x; y ) (x; y ) + max L 0 (
; t : A(A + 1)

0 (
;t)

© 2001 by Chapman & Hall/CRC



Z

(x; y) jk0 jds + maxL 0 (
; t)

Z

)

1  (x; y ) A(A + 1) 0

(
;t)

(x; y )k0 ds



(x+ ; y+ ; + ) + (x

i

)

7.3.

195

Shrinking to a point

An appli ation of the mean value theorem shows d log G dt t=t

> L2 (x+; y+)k+0 + (x +

C4   A(A + 1)

( +

Finally, we use (7.11), (7.12), and + 1 n d C log G > dt t=t A(A + 1) 4

; y )k0



 )




 C5 1 + +

L





C5 L 0 (
; t) : A(A + 1)

 = jk0 jL to get o


 C6 g(t) + L 0 (
; t) + A ; (7.14)

where C4 and C6 depend only on ; , and the initial urve. Observe that 0 (
; t) onverges to a portion of  smoothly away from Q and the total absolute urvature of 0 (
; t) is uniformly bounded. By repla ing 0 (
; t) by the portion lying on a suÆ iently small neigh
borhood of the origin, if ne essary, we may assume L 0 (
; t) and A are so small that  
1 C6 L 0 (
; t) + A < C4  ; for all t 2 (t1 ; !) : 2 Hen e, by (7.14), for t < t, and lose to t, g(t) 6 G( t ) < G( t ) = g(t) :

We on lude that there exists Æ 6 "0 su h that g(t) > Æ > 0 for all t in [t1 ; !).  Let (
; t) be the solution (7.1) where (7.4) holds. If ! is nite, (
; t) shrinks to a point as t " !.

Theorem 7.8

We employ the blow-up argument in x5.2 to (7.10). In ase the ow does not shrink to a point, we get a limit ow 1 satisfying

Proof:

 1 = (0; 0)k1 n0 : t

© 2001 by Chapman & Hall/CRC

196

CH. 7.

Embedded Closed Geodesi s on Surfa es

As before, one an show that 1 is uniformly onvex and with total absolute urvature . Now we an follow the proof of Theorem 5.1, using Lemma 7.7 to repla e Proposition 5.2, to draw a ontradi tion.



7.4

Convergen e to a geodesi

In this se tion, we prove the following theorem, whi h omplements Theorem 7.8. Theorem 7.9

Let

t > 0,

If it exists for all

t

! 1.

(
; t)

(
; t)

sub onverges to an embedded, losed

(M; g).

Lemma 7.10

t Proof:

its urvature onverges to zero uniformly as

Consequently,

geodesi of

be the solution of (7.1) where (7.4) holds.

lim

Z

!1

k 2 (s; t)ds = 0 :

(
;t)

By Lemma 7.1, we have

d dt

Z

k 2 ds =

=

Z h

2k Fss + k2 F + KF

Z

( 2aks2




i

F k 3 ds

(7.15)

2as kks + ak4 + 2aKk2 )ds :

Here and in the followings, the integration is over (
; t). Sin e the ow exists for all time, its length L(t) must have a positive lower bound, for otherwise (
; t) would be ontained in a very small geodesi disk and, by (7.10), shrinks to a point in nite time. Consequently, we an nd some onstant C depending only on a and M su h that inf k (
; t)  C for all t. So, max k2 6 C 2 + L(t)

(
;t)

© 2001 by Chapman & Hall/CRC

Z

ks2 ds :

7.4.

197

Convergen e to a geodesi

Putting this into (7.15), we have d dt

Z

k ds 6 2

Z

aks ds + L(t) 2

Z

ks ds 2

Z

ak ds + C1 2

Z

k 2 ds :

By Lemma 7.1, for any " less than amin(2L(0)amax ) 1 , we an nd t1 su h that Z

k 2 ds <

(
;t1 )

"

(7.16)

2

and Z1 Z

(7.17)

ak 2 < "2 :

t1 (
;t)

Therefore, as long as the total squared urvature of (
; t) is less than ", d dt

Z

(
;t)

Z

amin

k ds 6 2

2

(
;t)

ks ds + C2 2

Z

k 2 ds

(7.18)

(
;t)

for some C2 . We laim that Z

k2 < "

(
;t)

for all t > t1 . For, if on the ontrary, there is a rst time t2 > t1 su h that Z

(
;t2 )

© 2001 by Chapman & Hall/CRC

k2 = " ;

198

CH. 7.

Embedded Closed Geodesi s on Surfa es

then, by (7.18), (7.16), and (7.17),

Z

Z

" 2

k 2 ds

k 2 ds

<

(
;t1 )

(
;t2 )

Z t2 Z

6

C2 amin t1


>
> :

(
; 0) = 0 :

By Proposition 1.2, (8.2) has a maximal solution in [0; !Æ ). By (1.16), the urvature of Æ , kÆ , evolves by kÆ = (FÆ )ss + kÆ2 FÆ : t

(8.3)

Let M0 be the supremum of the urvature of 0 . It follows from applying the maximum prin iple to (8.3) that there exist positive

onstants M and T depending only on M0 su h that kÆ (
; t)j 6 M ;

j

Lemma 8.2 Let

M0

 2 (0; 1).

and the length of

Z

Æ (
;t)


2

0

FÆ (kÆ ) s ds 6

© 2001 by Chapman & Hall/CRC

t 2 [0; T ℄ ;

8

Æ 2 (0; 1) :

(8.4)

C

depending on

8

There exists a onstant

su h that

C ; t 2 [0; T ℄ ; 0 < Æ < 1 : t

8.1.

205

Short time existen e

For simpli ity, we set

Proof:

k = kÆ ; v

=

F

vt

=

F 0 vss + F 0 k 2 v

wt

= (F

=

F Æ (k Æ ) ;

and

w = vs

=

FÆ0 (kÆ )(kÆ )s :

We have

and

00 0 ws )s + F k2 v + 3kv + F 0 k2
w : F0

We note from (8.4) that (F 0 )

1

F 00 k 2 v

+ 3kv +

F 0 k2

is uniformly

bounded on [0; T ℄. Therefore,

d 2 dt

1

Z

Z

Æ (
;t)

w ds 2

=

Æ

Z

6

Æ

1

and

B1

Z

vs ds 2

Æ

1

6

Z

Æ

Z

B2 L(0)

ws2 ds + B1

M0 .

2

Z


1

Æ

2

Æ

Æ

Æ

w2 ds w2 ds ;

By interpolation,

Z


1

v ds 2

Z

F 0 ws2 ds + B1

Æ

depend only on

6

w2 F k )ds

Z

6 where

(wwt

2 vss ds

ws2 ds


1


1 2

2

;

where L(0) is the initial length. Therefore, letting 2 =

d 2 dt 1

Z

w ds 6 2

Æ

© 2001 by Chapman & Hall/CRC

Z

2

w ds 2

Æ


2

1 (L(0)B22 )

Z + B1

Æ

w2 ds ;

1

,

206

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

whi h implies

Z

Æ

w2 ds 6 max

 1

2 t

;

2B1

2

:



Lemma 8.1 implies that the urvature of Æ (
; t) is uniformly Holder ontinuous on every ompa t subset of S 1  (0; T ℄. Thus, we

an nd a subsequen e f Æj (
; t)g whi h C 2 - onverges to a solution of (8.1) in every ompa t subset of S 1  (0; T ℄. Next, by applying the weak omparison prin iple (x1.2) to (1.5), where F is given by (8.1), we have the ontainment prin iple: let

1 and 2 be two GCSFs in [0; T ). Suppose that 1 (
; 0) is bounded by 2 (
; 0). Then, 1 (
; t) is bounded by 2 (
; t) for all t 2 (0; T ). As a

onsequen e of this prin iple, we obtain the existen e and uniqueness of (8.1) as well as the niteness of !. Summing up, we have the following proposition. For every C 2 -embedded losed 0 , the Cau hy problem for the GCSF has a unique, embedded solution in C (S 1  [0; !)) \ Ce2; (S 1  (0; !)) for some  2 (0; 1) and ! is nite. The urvature be omes unbounded as t " !. Proposition 8.3

It remains to show that the ow is embedded. For uniformly paraboli ows, the embeddedness preserving property is a onsequen e of the strong maximum prin iple. Although the strong maximum prin iple is not available now, we an still prove that the ow is embedded by the monotoni ity property of the following isoperimetri ratio introdu ed by Huisken [78℄. In fa t, we shall see more appli ations of this ratio in the subsequent se tions. Let : [a; b℄  [t0 ; t1 ℄ ! R2 be an evolving ar satisfying (8.1) . De ne the extrinsi and intrinsi distan e fun tions d and ` : [a; b℄2 

© 2001 by Chapman & Hall/CRC

8.1.

207

Short time existen e

[t0 ; t1 ℄

! R by d(p; q; t) = j (p; t)

(q; t)j

and Z

`(p; q; t) = Proposition 8.4

(a; b)2

p

q

ds(
; t) :

Suppose d=` attains a lo al minimum at (p; q; t) 2

 (t0; t1 ). Then,

d d ( )(p; q; t) > 0 ; dt `

(8.5)

and \=" holds if and only if is a straight line. Consequently, inf(d=`)

is in reasing as long as it attains interior minimum.

Proof:

Sin e

d=`

we may assume

has a global maximum on the diagonal of [a; b℄2 ,

p 6= q

and

s(p) > s(q )

at t.

Denote

!=

(q; t)

j (q; t)

(p; t)

(p; t)j

and

d

(p; t) ; e1 = ds

and

d

(q; t) : e2 = ds

By the assumption, we have 0

=

=

0

=

=

© 2001 by Chapman & Hall/CRC

d  j (s(p) + s; t) (q; t)j  ds s=0 `+s 1

`

h ! ; e1 i

d ; `2

(8.6)

d  j (p; t) (s(q ) + s; t)j  ds s=0 ` s 1

`

h!; e2 i + `d2

;

(8.7)

208

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

and

0

6

d2  j (s(p) + s; t) (s(q ) ds2 s=0 ` + 2s

=

2 `2

s; t)j 

h 2

h!; e1 + e2i + 1` je1 +d e2j + !; k(q; t)n(q; t)

k (p; t)n(p; t)



h!; e1 + e2i2 i + 4d : `3

d

By (8.6) and (8.7), 2 `2 Noti ing that

h!; e1 + e2i + 4`d3 = 0 :

! ==e1 + e2 , we have

je1 + e2j2 = h!; e1 + e2i2 : Thus,

0



6 !; k(q; t)n(q; t)



k (p; t)n(p; t) :

(8.8)

By (8.6) and (8.7), again,





!; n(q; t)

=

=

p



!; n(p; t)

1

(d=`)2

>0:

So,

k (q; t) + k (p; t) > 0 :

© 2001 by Chapman & Hall/CRC

(8.9)

8.1.

209

Short time existen e

Now, d d
(p; q; t) dt `

1

=

>

`


d d2
21  1  1 j k j k ( q; t ) + j k j k ( p; t ) + 2 2 ` `

1

Z q

p

jkj+1 ds(
; t) 

0:

More generally, Proposition 8.4 holds for the ow (1.2) if F = F (q) is odd and paraboli .

Remark 8.5

In on luding this se tion, we state two maximum prin iples for the urvature and the tangent angle of the ow. On where the urvature does not vanish we have, by (1.15) and (1.16), kt = (jk j 1 k )ss + k 2 (jk j 1 k )

(8.10)

t =  jk j 1 ss

(8.11)

and

k (p; t0 )

>

0

Lemma 8.6

(a) If

k (b; t0 ) > 0,

then, for some small

(t0 ; t0 + "). (b) If

 (p; t0 ) > 0 ,

then, for some small

for

p 2 [a; b℄

2

[a; b℄ and k(a; t0 ) > 0, " > 0, k (p; t) > 0 in [a; b℄ 

for

p

and

" > 0,  (p; t) > 0

 (a; t0 ) > 0 ,  (b; t0 ) > 0 ,

in

[a; b℄  (t0 ; t0 + ").

(a) and (b) an be proved in a similar way. In the following, we present a proof of (a). First we note that the ow an be approximated by a sequen e f Æ (
; t)g; Æ = Æj ! 0, satisfying (8.2). Its speed vÆ = FÆ (k) satis es

Proof:

vÆ 2v = FÆ0 (k) 2Æ + FÆ0 (k)k2 vÆ : t s

© 2001 by Chapman & Hall/CRC

(8.12)

210

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

Sin e the urvature of Æ is uniformly bounded, we an nd onstants  and M , 0 <  < 1 6 M , su h that FÆ (k) >  > 0 and 0 6 k2 FÆ (k) 6 M 0

0

for all Æ = Æj . We shall onstru t a subsolution of (8.12). Let sÆ (p; t) be the ar -length along Æ (
; t) from a to p. By the ommutation relation [=t; =s℄ = kvÆ =s, we have



 sÆ s t



= kvÆ ;

whi h is uniformly bounded. Set wÆ (p; t) = (sÆ (p; t) +  (t

t0 ) + 1; (t

t0 )) ;

where  and are onstants to be spe i ed later and (x;  ) =

1



p exp

x2 4



is the heat kernel. Noti e that x < 0 for x > 0 and xx > 0 for (2 )1=2 . Choose  > inf fsÆ =t : (p; t) 2 [a; b℄  [t0 ; t0 + "℄g where " > 0 is so small that (1 + ")2 > 2". It follows that (sÆ (p; t) +  (t t0 ) + 1)2 and

> (1 + ")2 > 2(t t0) 



wÆ s =   + + Æ x t t  s  2  wÆ Æ =  2 + + x s t 2 6 FÆ (k) sw2Æ + FÆ (k)k2 wÆ : 0

© 2001 by Chapman & Hall/CRC

0

8.2.

The number of onvex ar s

211

We on lude that wÆ is a subsolution of (8.12) in [a; b℄  [t0 ; t0 + ") for all Æj , and wÆ vanishes identi ally at t = t0 . If we further hoose " and small (if ne essary), we may assume that wÆ 6 vÆ on the lateral boundary of [a; b℄  [t0 ; t0 + "℄. By the maximum prin iple, wÆ 6 vÆ in [a; b℄  [t0 ; t0 +"℄. Letting j ! 1, we on lude v > w0 > 0 in [a; b℄  [t0 ; t0 + "℄.  8.2

The number of onvex ar s

Let (
; t) be a maximal solution of (8.1) in (0; !). An ar of (
; t) is alled a onvex ar if it is the image of a maximal ar [a; b℄  S 1 on whi h k(
; t) > 0 and k(p; t) > 0 for some p 2 (a; b). An ar is alled a on ave ar if it is the image of (a; b)  S 1 on whi h S k(
; t) 6 0 and k(p; t) < 0 for p 2 (a; a + ") (b "; b) for some " > 0. First, we show that the number of onvexn on ave ar s does not in rease in time. (p0 ; t0 ) with k(p0 ; t0 ) 6= 0, there exists a onp : [0; t0 ℄ ! S 1 su h that p(t0 ) = p0 and k(p(t); t) 6= 0

Lemma 8.7 For any tinuous map: for all

t.

Without loss of generality, we assume k(p0 ; t0 ) > 0. Consider the set f(p; t) 2 S 1  [0; t0 ℄ : k(p; t) > 0g and let U be its onne ted T

omponent ontaining (p0 ; t0 ). It is suÆ ient to show that U (S 1  f0g) is non-empty. Suppose this is not true. Then e t k has a positive maximum at (p1 ; t1 ) in U where t1 > 0. If we hoose > max k+1 , it follows U from (8.10) that 0 6 (e t k)t 6 (k+1 )e t k < 0 at (p1 ; t1 ). But this is impossible.  >From this lemma, we immediately on lude that, if 0 is omposed of nitely onvex and on ave ar s, then the number of onvexn on ave ar s does not in rease in time.

Proof:

© 2001 by Chapman & Hall/CRC

212

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

Next, we present a rather suprising feature for (8.1) , (0 <  < 1), due to Angenet-Sapiro-Tannenbaum [19℄. Proposition 8.8

Let

K

the number of onvex ar s of any solution of (8.1) , time

t

does not ex eed

K max where

0 . Then, 0 <  < 1, at

be the total absolute urvature of

C

n 1 + CL2 (0) t

1

1



;

2o ; 

(8.13)

is an absolute onstant.

When  = 1, we know that the ow be omes analyti at an instant, and so it has nitely many onvex and on ave ar s. However, no a priori estimate like (8.13) is known. By approximating the initial urve by analyti urves with nite in e tion points, it suÆ es to derive (8.13) assuming there are nitely many onvexn on ave ar s in [0; !). Let = f (p; t ) : p 6 p 6 p+ g be a onvex ar . There are intervals (p "; p ) and (p+ ; p+ + ") on whi h k(
; t ) < 0. Set U = the onne ted omponent of fk 6= 0g

ontaining the segment (p

"; p )  ft g ;

and U+ = the onne ted omponent of fk 6= 0g ontaining the segment (p+ ; p+ + ")  ft g.

By Lemma 8.7, for ea h t 2 [0; t ℄, there exist (p; t) 2 U and (q; t) 2 U+ . So, we an de ne p (t) = supfp : (p; t) 2 U

g p+(t) = inf fp : (p; t) 2 U+ g :

© 2001 by Chapman & Hall/CRC

and

8.2.

213

The number of onvex ar s

Let (
; t) be the image of [p (t); p+ (t)℄ under (
; t). We all (
; t); t 2 [0; t ℄ the history of the ar = (
; t ). Set q (t) = inf fp > p (t) : k(p; t) > 0g

q+(t) = supfp < p+ (t) : k(p; t) > 0g : Lemma 8.9

k(p; t) = 0

for all

and

S

p 2 [p (t); q (t)℄ [q+ (t); p+ (t)℄.

Suppose k(p0 ; t0 ) < 0 for some (p0 ; t0 ), p0 2 (p (t0 ); q (t0 )). By Lemma 8.6, k < 0 in [p (t0 ) "; p0 ℄  (t0 ; t0 + ") for small " > 0. We an onne t (p0 ; t0 ) to (p (t0 ) "; t0 ) in fk 6= 0g. In other words, (p0 ; t0 ) belongs to U . The ontradi tion holds. A similar argument shows that k vanishes on [q+ (t); p+ (t)℄.  Proof:

Lemma 8.10 We have

lim inf p (t) > p (t0 ) and lim sup p (t) 6 q (t0 ) ; t!t0 t!t0 and

lim sup p+ (t) 6 p+ (t0 ) and lim inf p+(t) > q+ (t0 ) : t!t0 t!t0 We only prove the rst two ases. For small " > 0, (p (t0 ) "; t0 )  U . Thus,

Proof:

k(p (t0 )

"; t) < 0;

8t 2 (t0

Æ; t0 + Æ) ;

where Æ is a small number. Sin e p (t) > p (t0 ) " for all t 2 (t0 Æ; t0 + Æ), by the de nition of p (t), the rst inequality holds. Next, by the de nition of q , k(q (t0 )+ "; t0 ) > 0 for some small ". By ontinuity, k(q (t0 ) + "; t) > 0 for t in jt t0 j < Æ, Æ small. We

laim that p (t) 6 q (t0 ) + " for all t 2 (t0 Æ; t0 + Æ). Let be the segment fq (t0 ) + "g  (t0 Æ; t0 + Æ). Suppose

© 2001 by Chapman & Hall/CRC

214

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

that p (t1 ) > q (t0 ) + " for some t1 2 (t0 Æ; t0 + Æ). There is some p1 2 (q (t0 ) + "; p (t1 )) su h that (p1 ; t1 ) 2 U . Applying Lemma 8.7 to [t1 ; t ℄, we an onne t (p1 ; t1 ) to (p (t ) "; t ) by a path T(S 1  [t ; t )). There is also a ontinuous fun tion u = 1 in U 1  T(S 1  (2) p (t); 0 6 t 6 t1 , u1 = p(2) (t1 ), whose graph 2 lies in U [0; t℄). The paths 1 and 2 together form a path in U onne ting (p (t ) "; t ) to the bottom S 1  f0g. Sin e k > 0 on , this path must be disjoint from . Consequently, p > q (t0 ) + " for any point (p; t) in this path for t 2 (t0 Æ; t0 + Æ). But this is in on i t with the de nition of p (t0 ) and q (t0 ).  Set t = f(p; t) : p (t) 6 p 6 p+ (t)g : Lemma 8.11

t

is stri tly nesting in time.

Let  (t) = (p (t); t) and + (t) = (p+ (t); t). We are going to show that  (t) is stri tly in reasing and + (t) stri tly de reasing. Sin e k(
; t) = 0 on [p (t); q (t)℄;  (t) = (p; t) for all p 2 [p (t); q (t)℄. Together with Lemma 8.10, we know that  (t) is ontinuous. At any xed t0 2 (0; t ), there is " > 0 su h that k(
; t0 ) < 0 on [p (t0 ) "; p (t0 )) and k(
; t0 ) > 0 on (q (t0 ); q (t0 ) + "℄. Thus,  (p; t0 ) >  (t0 ) on [p (t0 ) "; q (t0 ) + "℄ and the inequality is stri t at the endpoints of this interval. By Lemma 8.6, we have  (p; t) >  (t0 ) on [p (t0 ) "; q (t0 ) + "℄  (t0 ; t0 + Æ ) for some small Æ > 0. By Lemma 8.10, p (t) lies between p (t0 ) " and q (t0 ) + " for t lose to t0 . So,  (t) >  (t0 ) for t 2 (t0 ; t0 + Æ). We have shown that  (t) is stri tly in reasing in t. Similarly, one an show that + (t) is stri tly de reasing.  Now we an prove Proposition 8.8. Proof:

© 2001 by Chapman & Hall/CRC

8.2.

215

The number of onvex ar s

Let

4(t) = supfj(p1; t)

(p2; t)j : p (t) 6 p1 ; p2 6 p+ (t)g :

We rst show that

8 9 < t1 1  = 4(0) > min : 1 + CL(0)2 ; 2 ;

(8.14)

for some onstant C . Assume that 4(0) < =2. We let 0  [ 4(0)=2; 4(0)=2℄. On any stri tly onvexn on ave part of t , we an use the tangent angle to parametrize the ow. The resulting equation for v = jkj 1 k is given by v t

Let

= jvj 1+ (v + v) :

v = A(t) os

where



A(t) = A(0)

(8.15)

 3  44(0) ;

1+ 

We have

+ (4Ct (0))2





1+

:

 3  = 1 +  A (t ) 44(0)  +1       1 1+  3  3  2 1+ +1  2

os 44(0) > A(t) (4(0)) (4(0))2 4 v t



1+ +1 

C (4(0))2 os

= jv j 1+ (v  + v)

, provided C is suÆ iently small. So v is a supersolution of (8.15). By the maximum prin iple, we obtain, 2 C (4(0)) 1+ (8.16) jv(; t)j 6 

t 1+

© 2001 by Chapman & Hall/CRC

216

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

( hoose A(0) = 1) , and

jv(; t)j 6



C

1+



1+ Ct + sup jvj  2 (4(0)) 0





1+

sup jvj 0

(8.17)

( hoose A(0) = C sup jvj and C = ( os 38 ) 1 :) To explain how the 0 maximum prin iple is appli able, we observe that ea h t is the union of a nite number of onvexn on ave ar s. On ea h ar , the tangent angle  is single-valued and takes values in ( 4(0)=2; 4(0)=2) by Lemma 8.11. This image interval hanges in time, but v always vanishes at its endpoints. On the other hand, v has a uniformly positive lower bound in [ 4(0)=2; 4(0)=2℄ for t 2 [0; t ℄. Hen e, we an apply the maximum prin iple. Set t



T 1 (sup jvj; 0

2

= 2

1+



4(0))

(4(0))2 1+ : C(sup jvj)  0

We have 1 sup jvj : 2 0

sup jvj 6

T1=2

By the same argument, we get sup

t+T1=2 (t)

for T 1 (t) = T 1 (sup jvj; 2

2

t

jvj 6 12 sup jvj ; t

4(t)).

We now de ne an in reasing sequen e ftj g1 j =0 indu tively by setting tj +1 = tj + T 1 (sup jvj ; 2

© 2001 by Chapman & Hall/CRC

tj

4(tj )) :

8.2.

217

The number of onvex ar s

We have 1 sup jvj : 2 tj

sup tj+1 jvj 6 Sin e

Z

4(tj ) 6

tj

(sup jvj) 
L(tj ) ;

6 we have tj +1

2

6

tj

6 By (8.16), t1 = lim tj j !1

6 6

t0 +

Choosing t0 = (4(0))1

1

tj

1+



C

2

1+

2

1+

t0 + ,

jkjds(
; tj )



C



C

L2 L2

L2

1+

2

sup jvj



tj



j sup 0

2

 X 1

CL2 (4(0)) 1





0

2

2(1  ) 1+



t01+





jvj

 j 1

1







:



sup jvj

1





t0

:

we dedu e

t1 6 (1 + CL2 )(4(0))1



:

As the ar = t is not at, we must have t1 > t . Hen e, 1

4(0) >

t1  : 1 + CL2

To nish the proof, we note that disjoint onvex ar s have disjoint histories. So, the number of onvex ar s at t does not ex eed K max

© 2001 by Chapman & Hall/CRC

 1 + CL2 t

1

1



2 ; 



;

218

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

where K is the total absolute urvature of the initial urve.  In view of Proposition 8.8, without loss of generality, we may assume that the numbers of onvex and on ave ar s of the ow are always onstant. We all an evolving ar an evolving onvex (or on ave) ar if, for ea h t 2 [0; ! ), (
; t) is a onvexn on ave ar and the family f (
; t0 ) : 0 6 t0 < tg is the history of (
; t). Now the ow an be de omposed into a nite union of evolving

onvexn on ave ar s. Noti e by Lemma 8.11 the spheri al images of evolving onvexn on ave ar s are stri tly nesting in time. 8.3

The limit urve

First of all, by omparing the ow with an evolving ir le, one an argue as in the proof of Proposition 1.10 that

(
; t)  N 1+1  ( 0 ) : Ct

As before, the urve (
; t) onverges to a limit urve  in the Hausdor metri . In this se tion, we shall show that  is either a single point or a line segment. In the previous hapters, we have seen that ounting the interse tion points of two evolving urves is very useful in understanding the behaviour of the ow near the singularity. We now onsider this issue for (8.1). Given two embedded but not ne essarily losed C 2 - urves 1 and

2 , we ount their interse tions by the following method: (i) if 1 and 2 interse t at a point transverally, then ount the number of interse tions in a small neighborhood of this point, to be 1; and (ii) if 1 and 2 do not interse t transversally at a point, we hoose a

oordinate system on a suitable small tubular neighborhood along one of the urves su h that this neighborhood ontains the point and the restri tions of the urves in this neighborhood are expressed as

© 2001 by Chapman & Hall/CRC

8.3.

219

The limit urve

graphs of two fun tions u1 and u2 . We ount the number of interse tions in the neighborhood to be the number of sign hanges of u1 u2 . The number of rossing of 1 and 2 is the sum of the number of these interse tions. By Lemma 8.7, whi h only uses the weak maximum prin iple, we have: Let 1 and 2 : [0; 1℄  [0; T ) ! R be two evolving ar s of (8.1) whi h satisfy Proposition 8.12

 1 (
; t)

\ (
; t) =  (
; t) \ (
; t) =  2

2

1

for all t 2 (0; T ). Then, the number of rossing of 1 (
; t) and 2 (
; t) is non-in reasing in t. We point out that this proposition does not assert the number of rossing is nite, or is de reased by tangen y of interse tion. Let (
; t) be a maximal solution of (8.1). There exists nitely many points fQ1 ;


; Qm g on the limit urve  su h that  n fQ1 ;


; Qm g onsists of C 2 -ar s. Away from Qj s, (
; t)

onverges to  in C 2 -norm. Theorem 8.13

We losely follow the notation and the proof of Theorem 6.4. As before, f t g, the push-forward of jk(p; t)jdp, are uniformly bounded Borel measures in R2 , by Lemma 8.11. We an sele t a sequen e tj " ! su h that f tj g onverges weakly to a limit measure , Proof:

K

K

K

K = K + X KiÆQ ;

PK

i

i

where = 0 at points, fQi g is ountable, and K1 > K2 >


> 0 with Ki bounded by the total absolute urvature of the initial

urve. Let m be the rst integer for whi h Km+1 < . Then, for any

© 2001 by Chapman & Hall/CRC

220

P

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

2  n fQ1 ;


; Qmg, there is an " > 0 su h that K (D"(P )) < .

After throwing away the rst few tj s, we may assume there exists > 0 su h that

K t (D"(P )) 6 

(8.18)



j

for all j . By passing to a subsequen e, if ne essary, we an nd a xed T number N su h that there are exa tly N omponents in (
; tj ) D" (P ) whi h also interse t D"=2 (P ), and these omponents are graphs of uniformly Lips hitz ontinuous fun tions yj;1(x);


; yj;N (x) over some xed interval on the x-axis. For simpli ity, let's take P to be the origin. We an nd ;  > 0 small su h that the two line segments ` = [ ;  ℄  fg lie inside D"=2 (P ) and are disjoint from

(
; t) for all t 2 [tj0 ; !) where tj0 is lose to !. Any line segment onne ting `+ and ` is a stationary solution of (8.1). By Proposition 8.12, the number of rossing of and (
; t) is nonin reasing in time. When the slope of is steeper than those of yj0 ;1 ;


; yj0 ;N ; (
; t) will have exa tly N rossings with for all T t 2 [tj0 ; !). So, (
; t) [ =2; =2℄  [ ; ℄ onsists of exa tly N many urves whi h are graphs of uniformly Lips hitz ontinuous fun tions y1 (x; t);


; yN (x; t) whose Lips hitz onstants only depend on . Next, we derive a uniform C 2 -bound for these fun tions in [ =2+ Æ; =2 Æ℄  [tj0 + Æ; !) for small Æ > 0. Denote by ' a xed travelling wave solution of unit speed of (8.1) whi h is non-negative and onvex over some interval ( a; a), a 6 1, and let v(x; t) = '

x

x

   +

1

t+y ;

where > 0 and (x; y) 2 R2 (see x2.1). Let y = yi for some i and (x0 ; t0 ) 2 [ =2 + Æ; =2 Æ℄  [tj0 + Æ; !). We an hoose x; y

© 2001 by Chapman & Hall/CRC

8.3.

221

The limit urve

depending on su h that y (x0 ; t0 ) = v (x0 ; t0 )

and yx (x0 ; t0 ) = vx (x0 ; t0 ) :

A

ordingly, we hoose small (depending on Æ and ) su h that jvx(x; tj0 )j is greater than the uniform gradient bound of y in [ =2; =2℄ [tj0 ; !) whenever jv(x; tj0 )j 6 . We also require that v(x; tj0 ) >  for jxj > =2. Under these onditions, the fun tion y(x; tj0 ) v(x; tj0 ) has exa tly two simple zeroes in [ =2; =2℄ and  y ( ; t)

2

 v ( ; t ) < 0 ;

8 t 2 [t 0 ; !) :

2

j

On the other hand, w = y v satis es a uniformly paraboli equation of the form wt = a(x; t)wxx + b(x; t)wx ;

be ause v is uniformly onvex. By the Strum os illation theorem, w(
; t0 ) has no other zero other than the one at x0 . So, yxx (x0 ; t0 ) 6

1

'xx

x

0

x



:

A similar argument establishes a lower bound for yxx . We have proved that yi (1 6 i 6 N ) are C 2 -uniformly bounded in [ =2 + Æ; =2 Æ ℄  [tj0 + Æ; ! ). Now we may apply a regularity result of DiBennedetto on porous medium equations (x4.15 in DiBenedetto [43℄) to the evolution equation of (yi )xx (see (1.3)) to on lude that ea h (yi )xx is equi ontinuous in [ =2 + 2Æ; =2 2Æ℄  [tj0 + 2Æ; !). Thus, yi onverges to  in C 2 -norm around P . 

© 2001 by Chapman & Hall/CRC

222

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

Next, we shall show that in fa t onsists of line segments. Suppose there is a non-in e tion point on the regular part of . We shall see how this will lead to a ontradi tion. 



Let P be a non-in e tion point. There exist "0 > 0 T and to 2 [0; !) su h that D" (P ) (
; t) is a onne ted ar for all t in [t0 ; !). Lemma 8.14

Near the point P (
; t) and are C 2 with non-zero urvature. Hen e, the lemma follows from the strong maximum prin iple. 

Proof:



Proposition 8.15

 onsists of, at most, a nite number of line

segments. Suppose on the ontrary there are two regular non-in e tion points P and Q on n fQ1 ;


; Qm g. By Theorem 8.13, there exist a; b 2 S 1 su h that (a; t) and (b; t) tend to P and Q, respe tively, as t " !. By Lemma 8.14, we an nd D" (P ) and D" (Q) su h T T that (
; t) D" (P ) and (
; t) D" (Q) ontain a single onne ted

omponent of (
; t) for all t lose to !. In parti ular, we have

Proof:



dist( (a; t); (
; t) n D" (P )) > Æ and dist( (b; t); (
; t) n D" (Q)) > Æ ; for some Æ. Let p(t) be given by k (p(t); t)j = max jk (
; t)j :

j

(
;t)

Sin e jk(p(t); t)j ! 1 as t " !, there exists ftj g su h that k (p; t)j 6 jk (p(tj ); tj )j;

j

© 2001 by Chapman & Hall/CRC

(p; t) 2 S 1  [0; tj ℄ :

8

(8.19)

8.3.

223

The limit urve

Without loss of generality, we may assume a < p(tj ) < b ;

8j :

(8.20)

We onsider the res aling of the blow-up sequen e f (
; tj )g as we did in Chapter 5 by setting

j (p; t) =



(p(tj ) + "j p; tj + "1+ j t) "j

where (p; t) 2 R  [ tj "j 1  ; (!

(p(tj ); tj )

;

tj ) "j 1  ) and

"j = jk(p(tj ); tj )j 1 : Ea h j satis es (8.1) with jkj (p; t)j 6 1 = jkj (0; 0)j: By the regularity result of DiBenedetto [43℄, we an hoose a subsequen e, still denoted by f j g, whi h onverges to some 1 in C 2 -norm in every

ompa t subset of R  ( 1; 0℄. The limit 1 solves (8.1) with urvature k1 satisfying jk1 j 6 jk1 (0; 0)j = 1. It is also lear that 1 is omplete and non ompa t. We laim that it is also onvex and embedded. The number of onvexn on ave ar s of 1 is nite, and is nonin reasing in time by Lemma 8.7. Therefore, we may assume that the number of onvex ar s is onstant for t 2 ( 1; M℄; M > 0. It suÆ es to show 1 (
; t) is onvex in ( 1; M℄. Supposing not, then we an nd p; q lying on two adja ent ar s of 1(
; M) su h that k1 (p; M) < 0; k1 (q; M) > 0, and p < q. Let's denote by the evolving on ave ar ontaining (p; M) and by + the evolving onvex ar ontaining (q; M). For ea h t 2 ( 1; M℄ we let  (t) be the tangent angle of the in e tion point (or spot) whi h separates and + . By Lemma 8.11,  (t) is stri tly in reasing in t. So,  ( M)

© 2001 by Chapman & Hall/CRC

 ( M

1) > 2"0 > 0

224

CH. 8.

for some

"0 .

The Non- onvex Generalized Curve Shortening Flow

In other words, the minimum tangent angle of the two

adja ent ar s in reases at least 2"0 as time

t

goes from

M

1 to

M.

j , j (p; M ) belongs to an evolving on ave ar (j ) (j ) of j (
; t) and j (q; M ) belongs to an evolving onvex ar + of

j (
; t). Though (j ) and +(j ) may not be adja ent, it is lear that, (j ) (j ) and + must for large j , the minimum tangent angle of both in rease at least "0 as time goes from M 1 to M . By Lemma 8.11, the total absolute urvature of j (
; t) must drop at least "0 during [ M 1; M ℄. By the invarian e of the total absolute urvature under s aling, the total urvature of (
; t) drops at least "0 during 1+ 1+ [tj + "j ( M 1); tj + "j ( M )℄, but this is impossible. So, 1 For large

must be onvex and, hen e, is embedded. Our next laim is that the total absolute urvature of

1

must

. In fa t, if it is less than  at some t0 6 0, we an represent 1 (
; t0 ) as the graph of some onvex fun tion u(x; t0 ) over the entire x-axis. By the same reasoning in the proof of Proposition 5.4, the whole family (
; t) an also be represented as graphs of onvex fun tions u(x; t) whose gradients are uniformly bounded. It follows from the following lemma that uxx  0, ontradi ting jk1 (0; 0)j = 1. be equal to

Lemma 8.16 Let

u : R  [0; 1) ! [0; 1) ut =

where

juxj

and

juxxj

Let

A

;

(8.21)

1 1+ ;

8(x; t) 2 [

2; 2℄

 [2; 1) :

fjux (x; t)j : (x; t) 2 R  [0; 1)g.

= sup

Without loss

u(0; 1) = 1. Let v be an expanding self-similar satisfying v (x) > u(x; 1) and sup jvx j = A + 1 (see

of generality, assume solution of (8.1)

1)

are uniformly bounded. Then,

juxx(x; t)j 6 Ct Proof:

uxx 1 (3 (1 + u2x ) 2

be a onvex solution of

© 2001 by Chapman & Hall/CRC

8.3.

225

The limit urve

x2.1).

By the omparison prin iple, 1 1 1+1  0 6 u(x; t) 6 t v((1 + )t 1+ x) 1+ in R  [1; 1). Let x 
1
 w(x; t) = ' + (t 1) +

(8.22)

be the travelling wave solution of (8.1) with speed . Re all that ' is an even, nonnegative onvex fun tion in ( ; ),  > 2, satisfying '(0) = 0 and

j'0 (x)j > A in ( ; 2℄ [ [2; ) : (8.23) Fix (x0 ; t0 ) where x0 2 [ 2; 2℄ and t0 > 2. By (8.22), 1 (8.24) 0 6 u(x0 ; t0 ) 6 Bt01+ ; where B = maxfv(x) : x 2 [ 4; 4℄g. Now we rst hoose the wave 1 1 speed = (2B )  t01+ . By (8.23) and (8.24), we an hoose  2 [ 2

2 ; 2 + 2 ℄ and < 0 su h that

u(x0 ; t0 ) = w(x0 ; t0 ) and

ux (x0 ; t0 ) = wx (x0 ; t0 ) : On the other hand, it is easy to see that the graphs of u and w interse t exa tly twi e at t = 1. By applying the Sturm os illation theorem to u w, we on lude that u w has, at most, two zeroes ( ounting multipli ity) for t > 1. It follows that x 
1 'xx 0 uxx (x0 ; t0 ) 6

6

© 2001 by Chapman & Hall/CRC

1



(2B )  sup 'xx (x) : x 2 [ 2 1

t01+



2

2



; 2 + 2 2 ℄

226

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

for all x0 2 [ 2; 2℄ and t0 > 81+ (2B )1+  ( 2) 1



(1+ )

,  = minf; 3g.

We have shown that 1 always has total absolute urvature . Consequently, for any " > 0, we an hoose two points on 1 (
; 0) su h that the ratio of the extrinsi distan e d" and the intrinsi distan e `" at these two points is less than ", i.e.,

d" =`" < " :

(8.25)

Re all the hoi e of a and b in (8.19) and (8.20). We onsider the extrinsi and intrinsi distan es in [a; b℄  [t0 ; !), where t0 is hosen so that (8.19) holds in [t0 ; !). We an nd some Æ1 > 0 su h that   d(a; q; t) : q 2 [a; b℄; t 2 [t0 ; !) > Æ1 inf `(a; q; t) and  d(p; b; t)  inf : p 2 [a; b℄; t 2 [t0 ; !) > Æ1 : `(p; b; t) By Proposition 8.4, (we hoose " < Æ1 )  d(p; q; t)  : p; q 2 [a; b℄; t 2 [t0 ; !) inf `(p; q; t)    > inf Æ1 ; min dd((p;p; q;q; tt0 )) : p; q 2 [a; b℄ : 0 On the other hand, by (8.25), this is impossible. The ontradi tion shows that every regular ar of  must be at. Proposition 8.15 is proved. Finally, we prove the main result of this se tion.

Any maximal solution (
; t) of (8.1) onverges to a point or a line segment in the Hausdor metri as t " !. Theorem 8.17

By what we have shown, we know that, for any small " > 0, there exists t0 lose to ! su h that, for all t 2 [t0 ; !); (
; t) is ontained in the "2 -neighborhood of  and the total absolute urvaS ture of (
; t) outside i D" (Qi ) is less than ". In parti ular, ea h Proof:

© 2001 by Chapman & Hall/CRC

8.3.

227

The limit urve

S

omponent of (
; t) n i D" (Qi ) is a graph over the orresponding S line segment in  n i D" (Qi ). By ounting the rossing number of S

(
; t) n i D" (Qi ) with suitable transversal line segments, we may asS sume that the number of graphs over a line segment in  n i D" (Qi ) is onstant for all t. Suppose on the ontrary that  has two line segments 1 and 2 whi h meet at Q with an angle not equal to zero or . It is not hard to see that we an nd a onne ted ar C (t0 ) of (
; t0 ) satisfying (i) C (t0 ) is ontained in Dr0 =2 (Q), where r0 = minfjQi i 6= j g; and

Qj )j

:

(ii) C (t0 ) n D" (Q) onsists of two omponents C1 (t0 ) and C2 (t0 ) so T that Ci (t0 ) is a graph of some fun tion over i (Dr0 =2 (Q) n D" (Q)) for i = 1; 2. We follow the evolution of C (t0 ) to obtain a onne ted evolving ar C (t) of (
; t) for all t 2 [t0 ; ! ) satisfying T (i)0 C (t) is ontained in Dr0 =2 (Q) and onverges to  Dr0 =2 (Q) in the Hausdor metri as t " !, and (ii)0 C (t) n D" (Q) onsists of two omponents C1 (t) and C2 (t) so T that Ci (t); i = 1; 2; is the graph over i (Dr0 =2 (Q) n D" (Q)) T T whi h onverges to i (Dr0 =2 (Q) D" (Q)) in C 2 -norm.

Let's examine d=` for two points on C (t). As before, by Proposition 8.4, we have   d(p; q; t) : p:q 2 C (t); t 2 [t0 ; !) inf `(p; q; t) > Æ0 > 0

T

for some Æ0 . On the other hand, C (t) onverges to  Dr0 =2 (Q) and Q is a singularity. By repeating the blow-up argument in the proof of Theorem 8.15, we have a ontradi tion. Hen e,  must be a single line segment or a single point. 

© 2001 by Chapman & Hall/CRC

228 8.4

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

Removal of interior singularities

In the last se tion, we have shown that  is a line segment. It may happen that there are some singularities lying on its interior. Now we prepare to show that this is impossible. First of all, we need to establish a whisker lemma for (8.1). In view of the dis ussion in previous se tions, we may assume the number of onvexn on ave ar s is onstant in time. Furthermore, their total absolute urvature is either greater than  or less than  Æ0 for some positive Æ0 . By the de nition of onvex and on ave ar s, we know that ea h on ave ar annot have interior in e tion points, and, for a onvex ar , its in e tion spots, if they exist, always lie at the ends. For any real number , we all P 2 (
; t) a -point if its tangent angle is (mod 2).

Let A1 be a -point on (
; t) for some t1 2 (0; !). Suppose it is not an in e tion point. Then, there exists a C 2 -family fAt g, t 2 [0; t1 ℄, of -points on (
; t) su h that At1 = A1 . Lemma 8.18

As A1 is not an in e tion point, the impli it fun tion theorem ensures that there exists a C 2-family of -points for t near t1 ; t 6 t1 . Let (t0 ; t1 ℄ be the largest interval on whi h the family is de ned. The point At will a

umulate at in e tion spots of (
; t0 ). However, as any in e tion spot is onne ted by a onvex ar and a on ave ar , it follows from Lemma 8.6 that there are no -points around the a

umulation point for t slightly above t0 , and ontradi tion holds.

Proof:



An ar b(t)  (
; t) is alled a -inward ar (resp. -outward ar ) if the following hold: (a) the ar b(t) has negative (resp. positive) urvature,

© 2001 by Chapman & Hall/CRC

8.4.

229

Removal of interior singularities

(b) the inward (resp. outward) pointing unit tangents to b(t) at the endpoints are both given by ( os ; sin ), and ( ) both endpoints are not in e tion points of (
; t). By this de nition, the endpoints of a -inward ar (resp. -outward ar ) are a -point and a ( + )-point, respe tively. We an use Lemma 8.18 to tra e ba k in time to form a ontinuous family of -inward ar s (resp. -outward ar s). Now we have the following lemma, whose proof is similar to that of the whisker lemma in x6.4. Lemma 8.19 (whisker lemma) There exists Æ

0 su h that, for any P b(t)  (
; t) with t 2 (0; !),

on

on a the

-inward

ar

> 0 depending only (or -outward ar ),

Æ-whisker,

`P;Æ; = fP + r( os ; sin ) : 0 6 r 6 Æg ; is disjoint from

(
; t) n b(t).

The following lemma tells us that any onvex or on ave ar of total absolute urvature less than  Æ0 disappears as t " !. Lemma 8.20 There exists

K >0

n

depending only on

0

su h that

the urvature on any onvex on ave ar with total absolute urvature less than



Æ0

is bounded between

K

and

K.

Let C(t) be an evolving onvexn on ave ar with total absolute urvature less than  Æ0 . By disposing the in e tion spots, we may assume C(t) is free of in e tion points and an be parametrized by its tangent angle . Then, k = k(; t) satis es

Proof:

kt = k2 (v + v) ;

(8.26)

where v = jkj 1 k(; t). Without loss of generality, we may assume the tangent range of C(0) is ontained in [Æ0 =2;  Æ0 =2℄. By Lemma

© 2001 by Chapman & Hall/CRC

230

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

8.11, the tangent range of C (t) lies inside the same interval for all t > 0. Equation (8.26) admits two stationary solutions given by k = (A(sin

Æ0 ) 2

1

sin )  ;  2 [0; 2℄ ; 1

where A = max jk(
; 0)j . It follows from the omparison prin iple

0 that

jk(; t)j 6 A  1



Æ sin 0 2



1



for all t 2 [0; !).



Now, suppose that  is a line segment [0; `℄ on the x-axis. Let (0; 0) = Q1 < Q2 <


< Qm = (`; 0) be its singularities. For any small 2" < r0 = minfdist (Qi ; Qj ) : i 6= j g, there exists t" su h that, for all t > t", (
; t) is ontained in the "2 -neighborhood of  . Furthermore, (
; t) onverges in C 2 -norm to  away from m [

D" (Qj ), and the number of onne ted ar s, whi h are now graphs

j =1

of

C 2 -fun tions

over the intervals of [0; `℄ n

m [ j =1

D" (Qj ), are onstant

in time. We are going to show that m = 2 by assuming there is an interior singularity Q2 , and then draw a ontradi tion. Imagine at some time lose to !, an ar enters D" (Q2 ) from the side of Q1 or Q3 . It would stay and wriggle inside D" (Q2 ) and then ome out either along the same side or the opposite side. A

ording to the dierent possibilities, we lassify the ar into two types. Spe i ally, a onne ted ar (t0 ) of (
; t0 ) is of type I if its total absolute urvature is greater than  Æ0 =2 and it onne ts a point on the verti al line fx = Q2 r0 =2g to a point on the verti al line fx = Q2 + r0 =2g su h that (t0 ) \ ([Q2 r0 =2; Q2 r0 =4℄  R ) and

(t0 ) \ ([Q2 + r0 =4; Q2 + r0 =2℄  R) are onne ted. A onne ted ar

(t0 ) is of type II if it onne ts a verti al point (i.e., -point with

© 2001 by Chapman & Hall/CRC

8.4.

Removal of interior singularities

231

= =2) P (t0 ) to a point R(t0 ) on the verti al line fx = Q2 + r0 =2g (or fx = Q2 r0 =2g) su h that the following hold:

(II1 ) (t0 ) \ ([Q2 + r0 =4; Q2 + r0 =2℄  R) (or (t0 )\ ([Q2 r0 =2; Q 2 r0 =4℄  R)) is onne ted, (II2 ) (t0 ) \ ([Q2 r0 =2; Q2 r0 =4℄  R) (or (t0 ) \ ([Q2 + r0 =4; Q2 + r0 =2℄  R) has exa tly two omponents b1 (t0 ) and b2 (t0 ). Here, b1 (t0 ) is the subar whi h is loser to P (t0 ) and b2 (t0 ) is loser R(t0 ), (II3 ) P (t0 ) is not an in e tion point, and it lies on a onvexn on ave ar (t0 ) whose total urvature is greater than , (II4 ) the total absolute urvature of the subar d1 (t0 ) onne ting P (t0 ) and b1 (t0 ) is nearly =2, (II5 ) the total absolute urvature of the subar d2 (t0 ) onne ting R(t0 ) and b2 (t0 ) is nearly 0 , and (II6 ) denoting the tangent ve tor of (t0 ) at P (t0 ) along the dire tion from P (t0 ) to R(t0 ) by e, the ray fre : r > 0g interse ts to d2 (t0 ).

© 2001 by Chapman & Hall/CRC

232

CH. 8.

© 2001 by Chapman & Hall/CRC

The Non- onvex Generalized Curve Shortening Flow

8.4.

233

Removal of interior singularities

Lemma 8.21 For any small su h that

(
; t0 )

">

0,

there exists some

t0

lose to

!

admits either a type I or a type II ar around an

interior singularity.

Let (t0 ) be an evolving onvexn onvex ar developing the singularity Q2 , and O(t0 ) a verti al point on (t0 ). We shall nd a type Intype II ar by tra ing (
; t0 ) starting at O(t0 ). Let's rst tra e along the urve from O(t0 ) in the ounter lo kwise dire tion. It is obvious that we an nd a onne ted ar 1 0 (t0 ) whi h onne ts O(t0 ) to a point G1 (t0 ) on the verti al line fx = Q2 r0 =2g (or fx = Q2 + r0 =2g) su h that 1 0 (t0 ) \ ([Q2 r0 =2; Q2 r2 =4℄  R) (or 1 0 (t0 ) \ ([Q2 + r0 =4, Q2 + r0 =2℄  R)) is onne ted and

1 0 (t0 ) \ ([Q2 + r0 =4, Q2 + r0 =2℄  R) =  (or 1 0 (t0 ) \ ([Q2 r0 =2, Q2 r0 =4℄R = ). By the same reasoning, along the lo kwise dire tion, we an nd 2 0 (t0 ) onne ting O(t0 ) and some point G2 (t) lying on fx = Q2 r0 =2g (or fx = Q2 +r0 =2g) su h that 2 0 (t0 )\([Q2 r0 =2, Q2 r0 =4℄  R) (or 2 0 (t0 ) \ ([Q2 + r0 =4, Q2 + r0 =2℄  R)) is onne ted and 2 0 (t0 ) \ ([Q2 + r0 =4, Q2 + r0 =2℄  R) =  (or 2 0 (t0 ) \ ([Q2 r0 =2, Q2 r0 =4℄  R) = ). If G1 (t0 ) and G2 (t0 ) lie on dierent verti al lines, the union of 1 0 (t0 ) and 2 0 (t0 ) forms a type I ar . If not, we

onsider Case (1) G1 (t0 ) and G2 (t0 ) lie on fx = Q2 r0 =2g, and Case (2) G(t0 ) lies on fx = Q2 + r0 =2g separately. Consider Case (1). When tra ing the urve in the ounter lo kwise dire tion from G1 (t0 ), it is lear that we an nd an ar

1 00 (t0 ) onne ting G1 (t0 ) to a point R(t0 ) on the verti al line fx = Q2 + r0 =2g su h that 1 00 (t0 ) \ ([Q2 + r0 =4, Q2 + r0 =2℄  R) is onne ted. Now we want to hoose a better verti al point on 1 0 (t0 ) [ 1 00 (t0 ) to repla e O(t0 ). We start at R(t0 ) and tra e ba k along 1 0 (t0 )[ 1 00 (t0 ). Sin e (
; t0 ) lies inside the "2 -neighborhood of  , we will go in and out of the disk D" (Q2 ). Then it goes into D" (Q1 ) and then returns to hit a point P 0 (t0 ) on D" (Q2 ). Sin e 1 0 (t0 ) and 1 00 (t0 ) are on-

Proof:

© 2001 by Chapman & Hall/CRC

234

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

ne ted in [Q2 + r0 =4, Q2 + r0 =2℄  R, the urve annot leave the disk to the right. This means that, as tra ing from the rst re-hit point P 0 (t0 ) along 1 0 (t0 ) [ 1 00 (t0 ), we arrive at a verti al point, P (t0 ), for the rst time. In the following, we show that the subar , denoted by

(t0 ), of 1 0 (t0 ) [ 1 00 (t0 ), whi h onne ts P (t0 ) to R(t0 ), is of type II. First of all, it follows from the de nition of (t0 ) that (II1 ) and (II2 ) are ful lled. The sum of the total absolute urvature of the \small" onvexn on ave ar is arbitrarily small when t0 is suÆ iently

lose to !. As we tra e from P 0 (t0 ) to P (t0 ), the tangent turns nearly =2, and so it must meet a \large" onvexn on ave (t0 ). Together with the fa t that P (t0 ) is the rst verti al point from P 0 (t0 ), we know that (II3 ) and (II4 ) also hold. Now, onsider the subar d2 (t0 ) onne ting R(t0 ) to b2 (t0 ). Suppose that the total absolute

urvature of d2 (t0 ) is not arbitrarily small. Then, d2 (t0 ) ontains a

onvexn on ave subar whose total urvature inside D" (Q2 ) is not less than  Æ0 =4. This says that d2 (t0 ) is of type I. So, we may always assume that (II5 ) holds in this ase. Finally, to verify (II6 ), we tra e the urve from b1 (t0 ) into D" (Q1 ) in ounter lo kwise dire tion. Sin e Q1 is the left endpoint of  , we will stop at a rst verti al point, P 00 (t0 ), in D" (Q1 ). The same reasoning as above shows that P 00 (t0 ) is not an in e tion point, and it lies on a onvexn on ave ar 00 (t ) whose total urvature is greater than  , and the total absolute 0

urvature of the ar d3 (t0 ) of (t0 ), whi h onne ts b1 (t0 ) to P 00 (t0 ), is nearly =2. It follows from the whisker lemma that 00 (t0 ) is a onvex ar , and the Æ-whisker at P 00 (t0 ), `P (t0 );Æ; ( is lose to ), is disjoint from (
; t0 )nfP 00 (t0 )g. Also, the Æ-whisker at P (t0 ), `P (t0 );Æ;0, is disjoint from (
; t0 )nfP (t0 )g. If the ar (t0 ) whi h ontains P (t0 ) is onvex, any point slightly passing beyond P (t0 ) in the lo kwise dire tion would not be able to be onne ted to G2(t0 ) by avoiding the Æ-whisker at P (t0 ). Thus, (t0 ) must be on ave. Furthermore, 00

© 2001 by Chapman & Hall/CRC

8.4.

235

Removal of interior singularities

the Æ-whisker at P (t0 ) for es (II6 ) to hold in this ase. Next, we onsider Case (2). Exa tly as before, we tra e along the urve starting at G1 (t0 ) in the ounter lo kwise dire tion to get an subar 1 (t0 ) onne ting G1 (t0 ) to some R1 (t0 ) on fx = Q2 r0 =2g so that 1 (t0 ) \ ([Q2 r2 =2; Q2 r0 =4℄  R ) is onne ted. Then, by tra king ba k from R1 (t0 ) along 1 (t0 ) [ 1 (t0 ), we nd a verti al point P1 (t0 ) on 1 (t0 ) [ 1 (t0 ) and a sub-ar

1 (t0 ) of 1 (t0 ) [ 1 (t0 ) onne ting P1 (t0 ) and R1 (t0 ) su h that (II1 ) { (II5 ) hold with, (t0 ); P (t0 ); R(t0 ); b1 (t0 ); b2 (t0 ); d1 (t0 ); d2 (t0 ) re(1) (1) (1) pla ed by 1 (t0 ); P1 (t0 ); R1 (t0 ); b(1) 1 (t0 ); b2 (t0 ); d1 (t0 ); d2 (t0 ), a

ordingly. Similarly, we tra e the urve starting at G2 (t0 ) along the lo kwise dire tion to get a onne ted subar 2 (t0 ) onne ting G2 (t0 ) to some R2 (t0 ) on the verti al line fx = Q2 r0 =2g. Then, we tra e ba k from R2 (t0 ) along 2 (t0 ) [ 2 (t0 ) to get a subar

2 (t0 ) of 2 (t0 ) [ 2 (t0 ) whi h onne ts a verti al point P2 (t0 ) to R1 (t0 ) su h that (II1 ) { (II5 ) hold with (t0 ); P (t0 )


; repla ed by

2 (t0 ); P2 (t0 )


; a

ordingly. The same as in Case (1), it follows from (II3 ), (II4 ), and the whisker lemma that the Æ-whisker at P1 (t0 ), `P1 (t0 );Æ; , is disjoint from

(
; t0 ) n fP1 (t0 )g and the Æ -whisker at P2 (t0 ), `P2 (t0 );Æ; , is disjoint from (
; t0 ) nfP2 (t0 )g. Also, as we ontinue to tra e along the urve from R1 (t0 ) to R2 (t0 ) to the left, we will arrive at a orresponding rst verti al point P1 (t0 ) and P2 (t0 ) in D" (Q1 ) su h that the Æwhisker at P1 (t0 ), `P1 (t0 );Æ;1 , where 1 is lose to , is disjoint from

(
; t0 ) nfP1 (t0 )g and the Æ -whisker at P2 (t0 ), `P2 (t0 );Æ;2 , where 2

lose to , is disjoint from (
; t0 ) n fP2 (t0 )g. Let's denote by e1 (or e2 ) the tangent of 1 (t0 ) (resp. 2 (t0 )) at P1 (t0 ) (resp. P2 (t0 )) along the dire tion from P1 (t0 ) (resp. P2 (t0 )) to R1 (t0 ) (resp. R2 (t0 )). We

laim that either 1 (t0 ) or 2 (t0 ) satis es (II6 ) For, suppose 1 (t0 ) does not satisfy (II6 ). In other words, the 00

00

00

0

0

0

00

00

00

0

0

00

00

00

00

00

00

00

00

00

© 2001 by Chapman & Hall/CRC

00

00

236

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

00 ray fre1 : r > 0g does not interse t d(1) 2 (t0 ). Note that P1 (t0 ) lies (1) (2) on a onvex ar . We have already seen that b(1) 1 (t0 ); b2 (t0 ); b1 (t0 ), and b(2) 2 (t0 ) are disjoint graphs over [Q2 + r0 =4; Q2 + r0 =2℄. The em(2) beddedness of the urve and the Æ-whiskers for e b(2) 1 (t0 ) and b2 (t0 ) (1) to lie between b(1) 1 (t0 ) and b2 (t0 ), and, hen e, 2 (t0 ) must satisfy (II6 ).  Now, we an show: Proposition 8.22 There are no interior singularities on

.

By Lemma 8.21, it suÆ es to show that there are no type In type II ar s on (
; t0 ) when t0 is lose to !. Suppose that (t0 ) is a type I ar . We follow the evolution of

(t0 ) to obtain an evolving ar (t) of (
; t), t 2 [t0 ; ! ), where (t)

onne ts a point P (t) on fx = Q2 r0 =2g to another point R(t) on fx = Q2 + r0 =2g su h that both (t) \ ([Q1 r0 =2; Q2 r0=4℄  R) and (t) \ ([Q2 + r0 =4; Q2 + r0 =2℄  R) are onne ted and ollapse to the x-axis in C 2-norm. There is a onvexn on ave subar of (t0 ) whose total urvature inside D" (Q2 ) is nearly . (Noti e that the whisker lemma for es that its total urvature annot be mu h away from .) So (t) ontinues to have total absolute urvature lose to  in D" (Q2 ) for all t 2 [t0 ; ! ) as it develops a singularity at Q2 . We look at the ratio of extrinsi and intrinsi distan es of two points on (t). Exa tly as before, by Proposition 8.4 and the C 2 onvergen e of the ow away from the singularities, we know that Proof:

inf

 d(P; Q; t) `(P; Q; t)

: P; Q 2 (t); t 2 [t0 ; !)



>Æ

0

>0

for some Æ0 . However, as (t) develops a singularity at Q2 , by repeating the blow-up argument in Proposition 8.15, we dedu e that inf

 d(P; Q; t) `(P; Q; t)

© 2001 by Chapman & Hall/CRC



: P; Q 2 (t); t 2 [t0 ; !) = 0 ;

8.4.

Removal of interior singularities

237

and the ontradi tion holds. Hen e, (
; t0 ) annot admit any type I ar . Next, suppose (t0 ) is a type II ar of (
; t). Without loss of generality, we may assume (t0 ) onne ts a verti al point P (t0 ) in D" (Q2 ) to some R(t0 ) on fx = Q2 + r0 =2g satisfying (II1 ) { (II6 ). Re all that P (t0 ) is a non-in e tion point on a onvexn on ave ar (t0 ) whose total urvature is greater than . Let (t) be the orresponding evolving onvexn on ave ar from (t0 ). By the whisker lemma, the total urvature of (t) tends to  and the unique verti al point P (t) is always a non-in e tion point. By the impli it fun tion theorem, P (t) is C 2 in t 2 [t0 ; !). We an follow the evolution of

(t0 ) to get a onne ted evolving ar (t) of (
; t) whi h onne ts P (t) to some R(t) on fx = Q2 + r0 =2g su h that for all t 2 [t0 ; ! ), (II1 ) (t) \ ([Q2 + r2 =4; Q2 + r0 =2℄  R) is onne ted, 0

(II2 ) (t) \ ([Q2 r0 =2; Q2 r0 =4℄  R) has exa tly two omponents b1 (t) and b2 (t), where b1 (t) is the ar loser to P (t), 0

(II3 ) P (t) is the only verti al point of (t), 0

(II4 ) the total absolute urvature of the subar d1 (t) of (t) whi h

onne ts P (t) and b1 (t) tends to =2, 0

(II5 ) the total absolute urvature of the subar d2 (t) of (t) whi h

onne ts R(t) and b2 (t) tends to 0, and 0

(II6 ) the verti al ve tor e, whi h is the tangent at P (t) along the dire tion from P (t) to R(t), points in su h a way that the ray fre : r > 0g interse ts d2 (t). 0

Here, in (II4 ) and (II5 ) , we have used C 2- onvergen e of the ow away from the singularities and the fa t that the spheri al image of ea h evolving onvexn on ave ar is stri tly nesting in time. As before, onsider the ratio of extrinsi and intrinsi distan es on (t). We laim that, for ea h t 2 [t0 ; !), the ratio attains its 0

© 2001 by Chapman & Hall/CRC

0

238

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

minimum in the interior of (t)  (t). For, the boundary of (t)  (t)

onsists of fP (t)g  (t) and (t)  fR(t)g. By the C 2 - onvergen e of (t) inside [Q2 + r0 =4; Q2 + r0 =2℄  R , it is lear that the minimum of d=` over fP (t)g  (t) is stri tly less than the its minimum over

(t)  fR(t)g when t0 is lose to ! . It is also lear that there is some interior point P (t) on (t) su h that d(P (t); P (t); t)=`(P (t); P (t); t) attains the minimum of d=` inside fP (t)g  (t). For simpli ity, let's assume the ar length parametrization satis es s(P (t)) > s(P (t)). Let 0

0

0

0

! = j

((PP ((tt));; tt))

(P 0 (t); t)

(P 0 (t); t)j

and

e

0

=

d

(P 0 (t); t) : ds

At (P (t); P (t); t), we have 0

d j (s(P 0 (t)) + s; t) (s(P (t)); t)j 0 = ds s=0 `+s

1

h !; e i `

=

0

d : `2

So, d : `

h!; e i = 0

(8.27)

By de nition, the right-hand side of (8.27) an be arbitrarily small as t0 tends to !. It follows from (II4 ), (II5 ) , and (8.27) that P (t) lies on the subar d2 (t), and ! approa hes e as t " !. We ompute 0

0



d ds s=0

=

1 `



j (s(P (t)) + s; t) `

d `

(h! ; ei + ) < 0 ;

© 2001 by Chapman & Hall/CRC

s

(s(P 0 (t)); t)j

0



8.5.

239

The almost onvexity theorem

as t0 " !. Thus, we an nd some interior point P 00 (t) with ar length parameter s(P (t)) + s0 (s0 > 0) su h that d(P 00 (t); P 0 (t); t) d(P (t); P 0 (t); t) < : `(P 00 (t); P 0 (t); t) `(P (t); P 0 (t); t)

So, d=` attains its minimum in the interior. By Proposition 8.4, inf

>

 d(P; Q; t) `(P; Q; t)



: P; Q 2 (t); t 2 [t0 ; !)

Æ0 > 0

for some positive Æ0 . On the other hand, as (t) develops a singularity at some Qj with j 6= 2, this in mum should be zero. This leads to a

ontradi tion. The proof of Proposition 8.22 is ompleted.  8.5

The almost onvexity theorem

Finally, we an prove the Theorem 8.1. In view of Theorem 8.17 and Proposition 8.22, it remains to show that there is no evolving on ave ar with total urvature greater than  at the endpoints for all t suÆ iently lose to !. Suppose that there is su h an evolving on ave ar (t) whi h persists at the end. We an nd a -inward ar b(t)  (t) ( = 0 or ), and so 

annot be a point. So, for all small " > 0, (
; t) n D" (Q1 ) [ D" (Q2 )

onsists of exa tly two ar s onverging to a line segment in C 2 -norm. Without loss of generality, we assume (t) develops a singularity at Q1 . Let G1 (t) and G2 (t) be two points of (
; t) lying on the middle line fx = `=2g and G1 (t) lies below G2 (t). To understand the behaviour of (
; t) inside D" (Q1 ), we tra e the urve from G1 (t) in the lo kwise dire tion. It goes into D" (Q1 ) and hits a rst verti al point P1 (t). Sin e the number of onvexn on ave ar s is xed and their total urvature either tends to 0 or is always greater than  as

© 2001 by Chapman & Hall/CRC

240

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

!, P1 (t) must be lo ated on a onvexn on ave ar 1 (t) whose total urvature is greater than , and the total absolute urvature of the ar onne ting G1 (t) and P1 (t) tends to =2. By the whisker lemma, 1 (t) must be onvex. We laim that the total urvature of 1 (t) tends to  . To see this, we tra e the urve from 1 (t) along the lo kwise t

"

dire tion. After passing through one or more \small onvexn on ave ar s," we will arrive at a on aven onvex ar 2 (t) whose total urvature is greater then . If the total urvature of 1 (t) is uniformly greater than , there is some -inward ar (or -outward ar ) b(t) on the se ond \large ar " 2 (t) su h that the orresponding Æ-whisker on b(t) rosses (
; t) n b(t). This ontradi ts the whisker lemma. Hen e, the total urvature of 1 (t) tends to  at the end. Let's examine the next \large ar ," 2 (t). The total absolute

urvature of the ar between 1 (t) and 2 (t) tends to zero. There exists a verti al point P2 (t) on 2 (t). Again, by de nition and the whisker lemma, there is a -inward ar (or -outward ar ) on 2 (t) with = 0, and the total urvature of 2 (t) also tends to . Observe that 2 (t) is on ave. In fa t, if it is onvex, the horizontal Æ-whisker `P2 (t);Æ;0 would prevent the points slightly beyond P2 (t) in the lo kwise dire tion from onne ting G2 (t) along (
; t). Continue to tra e along (
; t) in the lo kwise dire tion. After passing through some \small ar s" from 2 (t), we will get into the third \large ar ," 3 (t), with a verti al point P3 (t). It is lear that 3 (t) is onvex. If we do not meet any \large ar " by tra ing from 3 (t) to G2 (t), the total absolute urvature of the ar between 3 (t) and G2(t) is very small. So, the ar between 3 (t) and G2 (t) is nearly horizontal. The total

urvature of 3 (t) also tends to , and the total absolute urvature of the ar onne ting P3 (t) and G2 (t) tends to =2. If we meet a fourth \large ar ," 4 (t), then, exa tly as before, the total urvature

© 2001 by Chapman & Hall/CRC

8.5.

The almost onvexity theorem

241

of 3 (t) tends to , and 4 (t) is on ave with total urvature nearly equal to , too. By repeating this argument, we an de ompose the ar from G1 (t) to G2 (t) into \large" and \small ar s" where the \large ar s," 1 (t); 2 (t);


; 2n 1 (t); (n > 2), satisfy (i) the total

urvature of ea h i (t) tends to , and (ii) 2k 1 (t) is onvex and 1, there is a unique path 2k (t) is on ave. For ea h i; 1 6 i 6 2n Pi (t) onsisting of verti al points on i (t) whi h is C 2 for t 2 [t0 ; ! ): Consider (
; t) inside D" (Q1 ). Let R(t) 2 (
; t) \ D" (Q1 ) be the maximum point of jk(
; t)j on (
; t) \ D" (Q1 ). There exists a sequen e ftj g whi h onverges to ! su h that

jk(
; t)j 6 jk(R(tj ); tj )j; 8 t 6 tj ; on (
; t) \ D" (Q1 ). By Lemma 8.20, R(tj ) lies on some \large ar ." Without loss of generality, we may assume all R(tj ) lie on a single 1℄. By using a blow-up argument, we on lude i0 (tj ); i0 2 [1; 2n as before that  d(P; Q; t)  inf : P; Q 2 i0 (t); t 2 [t0 ; !) = 0 : (8.28) `(P; Q; t) Now we onsider two ases (a) i0 = 1 or 2n 1, and (b) i < i0 < 2n 1 separately. Take i0 = 1 in Case (a). Now we are in a similar situation of a type II ar near an interior singularity. Denote by 1 (t) the subar onne ting G1 (t) to the verti al point P2 (t) on 2 (t) along the lo kwise dire tion. As before, onsider the ratio d=` on 1 (t)  1 (t). It is lear that the minimum of d=` restri ted on the boundary of 1 (t)  1 (t) is attained at P2 (t) and some interior point P 0 (t) 2 1 (t). In the following, we assume the ar length parametrization is along the ounter lo kwise dire tion. Set 0 !1 = j

((PP2 ((tt));; tt))

((PP 0 ((tt));; tt))j ; 2

© 2001 by Chapman & Hall/CRC

242

CH. 8.

e2

=

d

(P2 (t); t) ds

e

=

d

(P 0 (t); t) : ds





0

We have

d 0 = ds s=0

1

=

`



1 `

and

j (s(P (t)) + s; t) 0

`+s

h!1; e i + d`

d ds s=0

=

The Non- onvex Generalized Curve Shortening Flow

0






(8.29)

`






;

j (s(P2 (t)) + s; t)

h!1; e2 i + d`

(s(P2 (t)); t)j

s

(s(P 0 (t)); t)j



(8.30)

:

In the following, we want to show that there exists a positive "0 su h that, whenever the minimum of d=` over 1 (t)  1 (t) is less than "0 , d=`, it attains its minimum in the interior of 1 (t)  1 (t). Let's assume that the minimum of d=` over 1 (t)  1 (t) is equal to the minimum over the boundary. By an algebrai argument, we may also assume the line segment between P2 (t) and P (t) has no interior interse tion with 1 (t). In fa t, suppose then is an interior interse tion Pe(t). We let d1 and `1 be the extrinsi and intrinsi distan es between P2 (t) and Pe(t), and let d2 and `2 be the extrinsi and intrinsi distan es between Pe(t) and P (t). Then d = d1 + d2 and ` = `1 + `2 . By de nition, 0

0

d1 + d2 `1 + `2

6 d`1 1

We have d2 `2

© 2001 by Chapman & Hall/CRC

6 d`1 1

;

:

8.5.

243

The almost onvexity theorem

and then

d2 `2

6 d`1 ++ `d2 ; 1

d=` at (Pe(t); P 0 (t)) is not greater than

whi h shows that the value of

d=`

the minimum of

2

over the boundary. Hen e, we may always as-

sume the interior of the line segment does not tou h 1 (t). When the minimum of small "0 ; (8.29) shows that to see that

P 0 (t )

rior of

e2 .

over

1 (t)  1 (t)

is less than a very

! 1 is nearly orthonormal to e . 0

annot lie on the ar between

avoid interior interse tion,

lose to

d=`

e

0

P1 (t)

and

P2 (t).

is nearly horizontal, and then

Thus, by (8.30),

1 (t)  1 (t).

d=`

!1

To is

attains its minimum in the inte-

Now we an apply Proposition 8.4 to on lude that positive lower bound in 1 (t)

It is easy

d=`

has a

 1 (t) for all t 2 [t0 ; !), a ontradi tion

to (8.28). Next, we treat Case (b). Assume that i0 = 2. We let the subar onne ting

P1 (t)

is lear that the minimum of some (P 00 (t); P3 (t)),

to

P3 (t)

d=`

over

P 00 (t) 2 2 (t).

be

in the lo kwise dire tion. It

 ( 2 (t)  2 (t))

is attained at

Set

!2

=

(P3 (t); t) j (P3 (t); t)

e1

=

d

(P1 (t); t) ; ds

e3

=

d

(P3 (t); t) ; ds

e

=

d

(P 00 (t); t) ; ds

00

2 (t)

(P 00 (t); t) ;

(P 00 (t); t)j

and

where the ar length parametrization is along the ounter lo kwise dire tion.

© 2001 by Chapman & Hall/CRC

244

CH. 8.

The Non- onvex Generalized Curve Shortening Flow

We rst laim that P (t) belongs to the interior of 2 (t). In fa t, if P (t) is an endpoint, then it must be P1 (t). The algebrai argument in the last paragraph shows that the line segment between P1 (t) and P3 (t) has no interior interse tion with 2 (t). Be ause P1 (t) and P3 (t) are verti al points, the line segment is verti al. However, by minimality, 00

00

0

6 = =



d ds s=0

1



j (s(P

1 d

(t))

s; t) (s(P3 (t)); t)j ` s






h!2 ; e1i + d`

`

00




1

` `

< 0;

whi h is impossible. Hen e, P (t) must be interior. As in Case (a), we want to show that the minimum of d=` over

2 (t)  2 (t) is attained in the interior whenever it is less than some small "0 . Again, we assume the minimum of d=` is attained on the boundary. The algebrai argument above shows that one may assume the line segment onne ting P3 (t) and P (t) has no interior interse tion with 2 (t). By minimality, we have 00

00



d 0 = ds s=0

=

1 `



j (s(P

00

(t)) + s; t) (s(P3 (t)); t)j `+s

h!2 ; e i + d` 00






(8.31)

:

When the minimum is less than some suitably small "0 , we see from (8.31) that !2 is nearly orthonormal to e . It is also lear that P (t) annot lie on the subar onne ting P3 (t) to P2 (t). To avoid an interior interse tion, e must be almost horizontal, and so !2 is 00

00

00

© 2001 by Chapman & Hall/CRC

8.5.

245

The almost onvexity theorem

lose to

e3 .

We ompute

d ds s=0

=

1 `



j (s(P3 (t)) + s; t)

h!2; e3i + d`

`

s

(s(P 00 (t)); t)j






< 0;

provided "0 is suitably small. So, d=` has an interior minimum and this leads to a ontradi tion as before. We have nally nished the proof of the Theorem 8.1. 

Notes Whether the Grayson onvexity theorem holds for the GCSF remains an open problem. In Angenent-Sapiro-Tannenbaum [19℄, it is proved that the aÆne CSF shrinks an embedded losed urve to a point with total absolute urvature onverging to 2. We have adapted and made straightforward generalizations of many results in this paper to (8.1) ,  2 (0; 1). Another main ingredient in the proof of Theorem 8.1 is the monotoni ity of Huisken's isoperimetri ratio [78℄. We observe that it ontinues to hold for a large lass of

ows, in luding (8.1) . See Remark 8.5.

© 2001 by Chapman & Hall/CRC

Bibliography This bibiography is not intended to be in lusive. For topi s not dire tly related to the ows dis ussed in this book, we usually list a few representative papers or a re ent work from whi h the reader an nd more referen es. [1℄ U. Abres h and J. Langer, The normalized urve shortening ow and homotheti solutions, J. Dierential Geom. 23 (1986), 175{196. [2℄ J. Ai, K.S. Chou and J. Wei, Self-similar solutions for the anisotropi aÆne

urve shortening problem, to appear in Cal . Var. PDEs. [3℄ S.J. Alts huler, Singularities of the urve shrinking ow for spa e urves, J. Dierential Geom. 34 (1991), 491{514. [4℄ S.J. Alts huler and M.A. Grayson, Shortening spa e urves and ow through singularities, J. Dierential Geom. 35 (1992), 283{298. [5℄ S.J. Alts huler and L.F. Wu, Convergen e to translating solutions for a

lass of quasilinear paraboli boundary problems, Math. Ann. 295 (1993), 761{765. [6℄ L. Alvarez, F. Gui hard, P.L. Lions and J.M. Morel, Axioms and fundamental equation of image pro essing, Ar h. Rational Me h. Ana. 132 (1993), 199{257. [7℄ L. Ambrosio and H.M. Soner, Level set approa h to mean urvature ow in arbitrary odimension, J. Dierential Geom. 43 (1996), 693{737. [8℄ B. Andrews, Contra tion of onvex hypersurfa es by their aÆne normal, J. Dierential Geom. 43 (1996), 207{230. [9℄ B. Andrews, Monotone quantities and unique limits for evolving onvex hypersurfa es, Inter. Math. Res. Noti es 20 (1997), 1001{1031. [10℄ B. Andrews, Evolving onvex urves, Cal . Var. PDEs 7 (1998), 315{371. [11℄ B. Andrews, The aÆne urve-lengthening ow, J. Reine Angrew. Math. 506 (1999), 43{83. [12℄ S.B. Angenent, The zero set of a solution of a paraboli equation, J. Reine Angrew. Math. 390 (1988), 79{96. [13℄ S.B. Angenent, Paraboli equations for urves on surfa es, I: Curves with p-integrable urvature, Ann. of Math. 132 (2) (1990), 451{483. 247 © 2001 by Chapman & Hall/CRC

248

Bibliography

[14℄ S.B. Angenent, Paraboli equations for urves on surfa es, II: Interse tions, blow-up and generalized solutions, Ann. of Math. (2) 133 (1991), 171{215. [15℄ S.B. Angenent, On the formation of singularities in the urve shortening

ow, J. Dierential Geom. 33 (1991), 601{633. [16℄ S.B. Angenent, In e tion points, extati points and urve shortening. Hamiltonian systems with three or more degrees of freedom (S'Agaro 1995), 3{10, NATO Adv. S i. Inst. Ser. C Math. Phys. S i. 553, Kluwer, Dordre ht, 1999. [17℄ S.B. Angenent and M.E. Gurtin, Multiphase thermome hani s with interfa ial stru ture, 2. Evolution of an isothermal interfa e, Ar h. Rational Me h. Anal. 108 (1989), 323{391. [18℄ S.B. Angenent and J.J.L. Velazquez, Asymptoti shape of usp singularities in urve shortening, Duke Math. J. 77 (1995), 71{110. [19℄ S.B. Angenent, G. Sapiro and A. Tannenbaum, On the aÆne heat equation for non- onvex urves, J. Amer. Math. So . 11 (1998), 601{634. [20℄ W. Ballman, G. Thorbergsson and W. Ziller, Existen e of losed geodesi s on positively urved manifolds, J. Dierential Geom. 18 (1983), 221{252.  [21℄ W. Blas hke, Uber AÆne Geometrie I: Isoperimetris he Eigens haften von Ellipse and Ellipsoid. Ber. Verh. Sa hs. Akad. Leipzig, Math. Phys. Kl. 68 (1916), 217{239. [22℄ K.A. Brakke, The motion of a surfa e by its mean urvature, Prin eton Univ. Press, Prin eton, N.J., 1978. [23℄ J.W. Cahn, C.M. Elliott and A. Novi k-Cohen, The Cahn-Hilliard equation with a on entration dependent mobility: motion by minus the Lapla ian of the mean urvature, European J. Appl. Math. 7 (1996), 287{301. [24℄ J.W. Cahn and J.E. Taylor, Surfa e motion by surfa e diusion, A ta Metallurgi a 42 (1994), 1045-1063. [25℄ E. Calabi, P.J. Olver and A. Tannenbaum, AÆne geometry, Curve ows and invariant numeri al approximations, Adv. Math. 124 (1996), 154{196. [26℄ S.Y.A. Chang, M.J. Gursky and P.C. Yang, The s alar urvature equation on 2- and 3-spheres, Cal . Var. PDEs 1 (1993), 205{229. [27℄ X. Chen, Generation and propagation of interfa es for rea tion-diusion equations, J. Dierential Eqn's. 96 (1992), 116{141. [28℄ Y.G. Chen, Y. Giga and S. Goto, Uniqueness and existen e of vis osity solutions of generalized mean urvature ow equations, J. Dierential Geom. 33 (1991), 749{786. [29℄ K.S. Chou (K. Tso), Deforming a hypersurfa e by its Guass-Kroue ker

urvature, Comm. Pure Appl. Math. 38, (1985), 867{882. [30℄ K.S. Chou and Y.C. Kwong, On quasilinear paraboli equations whi h admit global solutions for initial data with unrestri ted growth, to appear in Cal . Var. PDEs. [31℄ K.S. Chou and G. Li, Optimal systems of group invariant solutions for the generalized urve shortening ows, to appear in Comm. Anal. Geom.

© 2001 by Chapman & Hall/CRC

Bibliography

249

[32℄ K.S. Chou and X.J. Wang, A logarithmi Gauss urvature ow and the Minkowski problem, to appear in Analyse non lineaire. [33℄ K.S. Chou and L. Zhang, On the uniqueness of stable ultimate shapes for the anisotropi urve shortening problem, Manu. Math. 102(2000), 101{111. [34℄ K.S. Chou and X.P. Zhu, Shortening omplete plane urves, J. Dierential Geom. 50 (1998), 471{504. [35℄ K.S. Chou and X.P. Zhu, Anisotropi ows for onvex plane urves, Duke Math. J. 97 (1999), 579{619. [36℄ K.S. Chou and X.P. Zhu, A onvexity theorem for a lass of anisotropi

ows of plane urves, Indiana Univ. Math. J. 48 (1999), 139{154. [37℄ B. Chow, On Harna k's inequality and entropy for the Gaussian urvature

ow, Comm. Pure Appl. Math. 44 (1991), 469{483. [38℄ B. Chow and D.H. Tsai, Geometri expansion of onvex plane urves, J. Dierential Geom. 44 (1996), 312{330. [39℄ C. Croke, Poin are's problem and the length of the shortest losed geodesi on a onvex hypersurfa e, J. Dierential Geom. 17 (1982), 595{634. [40℄ P. DeMottoni and M. S hatzman, Development of interfa es in RN , Pro . Roy. So . Edinburgh Se t. A 116 (1990), 207{220. [41℄ K. De kelni k, Weak solution of the urve shortening ow, Cal . Var. PDEs 5 (1997), 489{510. [42℄ K. De kelni k, C.M. Elliott and G. Ri hardson, Long time asymptoti s for for ed urvature ow with appli ations to the motion of a super ondu ting vortex, Nonlinearity 10 (1997), 655{678. [43℄ E. DiBenedetto, Degenerate Paraboli Equations, Universitext series, Springer-Verlag, New York, (1993). [44℄ C. Dohmen and Y. Giga, Self-similar shrinking urves for anisotropi urvature ow equations, Pro . Japan A a. Ser. A 70 (1994), 252{255. [45℄ C. Dohmen, Y. Giga and N. Mizogu hi, Existen e of self-similar shrinking

urves for anisotropi urvature ow equations, Cal . Var. PDEs 4 (1996), 103{119. [46℄ K. E ker and G. Huisken, Mean urvature evolution of entire graphs, Ann. of Math. (2) 130 (1989), 453{471. [47℄ K. E ker and G. Huisken, Interior estimates for hypersurfa es moving by mean urvature, Invent. Math. 105 (1991), 547{569. [48℄ C.L. Epstein and M. Gage, The urve shortening ow, Wave Motion : Theory, Modeling and Computation, A Chorin and A Majda, Editors, Springer-Verlag, New York, 1987. [49℄ C.L. Epstein and M.I. Weinstein, A stable manifold theorem for the urve shortening equation, Comm. Pure Appl. Math. 40 (1987), 119{139. [50℄ L.C. Evans and J. Spru k, Motion of level sets by mean urvature I, J. Dierential Geom. 33 (1991), 635{681. [51℄ W.J. Firey, Shapes of worn stones, Mathemati a 21 (1974), 1{11.

© 2001 by Chapman & Hall/CRC

250

Bibliography

[52℄ M.L. Frankel and G.I. Sivashinsky, On the equation of a urved ame front, Phys. D 30 (1988), 28{42. [53℄ A. Friedman, Partial Dierential equations of Paraboli Type, Krieger, Malabar, FL, 1983. [54℄ M.E. Gage, An isoperimetri inequality with appli ations to urve shortening, Duke Math. J. 50 (1983), 1225{1229. [55℄ M.E. Gage, On an area-preserving evolution equation for plane urves, Nonlinear problems in geometry (Mobile, Ala., 1985), 51{62, Contemp. Math. 51, Amer. Math. So ., Providen e, R.L., 1986. [56℄ M.E. Gage, Deforming urves on onvex surfa es to simple losed geodesi s, Indiana Univ. Math. J. 39 (1990), 1037{1059. [57℄ M.E. Gage, Evolving plane uves by urvature in relative geometries, Duke Math. J. 72 (1993), 441{466. [58℄ M.E. Gage and R.S. Hamilton, The heat equation shrinking onvex plane

urves, J. Dierential Geom. 23 (1986), 69{96. [59℄ M.E. Gage and Y. Li, Evolving plane urves by urvature in relative geometries II, Duke Math. J. 75 (1994), 79-98. [60℄ C. Gerhardt, Flow of non onvex hypersurfa es into spheres, J. Dierential Geom. 32 (1990), 299{314. [61℄ M.H. Giga and Y. Giga, Evolving graphs by singular weighted urvature, Ar h. Rational Me h. Anal. 141 (1998), 117{198. [62℄ M.H. Giga and Y. Giga, Crystalline and level set ow{Convergen e of a

rystalline algorithm for a general anisotropi urvature ow in the plane, Gakuto International Series, Mathemati al S ien es and Appli ations Vol. 13 (2000), Free Boundary Problems: Theory and Appli ations, 64{79. (ed. N. Kenmo hi), Pro . of FBP'99. [63℄ Y. Giga, S. Goto, H. Ishii and M.H. Sato, Comparison prin iple and onvexity preserving properties for singular degenerate paraboli equations unbounded domains, Indiana Univ. Math. J. 40 (1991), 443{470. [64℄ Y. Giga and K. Ito, Loss of onvexity of simple losed urves moved by surfa e diusion, Topi s in nonlinear analysis, 305{320. Progr. Nonlinear Dierential Equations Appl. 35, Birkhauser, Basel 1999. [65℄ Y. Giga and R.V. Kohn, Asymptoti ally self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), 297{319. [66℄ M.A. Grayson, The heat equation shrinks embedded plane urves to round points, J. Dierential Geom. 26 (1987), 285{314. [67℄ M.A. Grayson, The shape of a gure-eight under the urve shortening

ow, Invent. Math. 96 (1989), 177{180. [68℄ M.A. Grayson, Shortening embedded urves, Ann. of Math. 129 (1989), 71{111. [69℄ R.E. Goldstein and D.M. Petri h, The Korteweg-de Vries hierar hy as dynami s of losed urves in the plane, Phys. Rev. Lett. 67 (1991), 3203{ 3206.

© 2001 by Chapman & Hall/CRC

Bibliography

251

[70℄ H.W. Guggenheimer, Dierential Geometry, M Graw-Hill, New York, 1963. [71℄ M.E. Gurtin, Thermome hani s of evolving phase boundaries in the plane, Oxford Mathemati al Mono., Clarendon, Oxford, 1993. [72℄ R.S. Hamilton, Isoperimetri estimates for the urve shrinking ow in the plane, Modern Methods in Complex Analysis (Prin eton, 1992), 201{222, Ann. of Math. Stud. 137, Prin eton Univ. Press, Prin eton, N.J. 1995. [73℄ R.S. Hamilton, The formation of singularities in the Ri

i ow, Surveys in dierential geometry Vol. II (Cambridge, MA, 1993), 7{136, Internat. Press, Cambridge, MA, 1995. [74℄ H. Hasimoto, A soliton on a vortex lament, J. Fluid Me h. 51 (1972), 477. [75℄ N. Hungerbuhler and K. Smo zyk, Soliton solutions for the mean urvature ow, preprint 1998. [76℄ G. Huisken, Non-parametri mean urvature evolution with boundary onditions, J. Di. Eqn's. 77 (1988), 369{378. [77℄ G. Huisken, Asymptoti behaviour for singularities of the mean urvature

ow, J. Dierential Geom. 31 (1990), 285{299. [78℄ G. Huisken, A distan e omparison prin iple for evolving urves, Asian J. Math. 2 (1998), 127{133. [79℄ G. Huisken and C. Sinestrari, Convexity estimates for mean urvature ow and singularities of mean onvex surfa es, A ta Math. 183 (1999), 45{70. [80℄ R. Ikota, N. Ishimura and T. Yamagu hi, On the stru ture of steady solutions for the kinemati model of spiral waves in ex itable media, Japan J. Indust. Appl. Math. 15 (1998), 317-330. [81℄ T. Ilmanen, Ellipti regularization and partial regularity for motion by mean urvature, Mem. Amer. Math. So . 108 (1994), no. 520. [82℄ J.P. Keener, Symmetri spirals in media with relaxation kineti s and two diusion spe ies, Phys. D. 70 (1994), 61{73. [83℄ J.P. Keener and J.J. Tyson, The dynami s of s roll waves in ex itable media, SIAM Rev. 34 (1992), 1{39. [84℄ J.P. Keener and J. Sneyd, Mathemati al Physiology, Springer-Verlag, NY 1998. [85℄ W. Klingenberg, Le tures on Closed Geodesi s, Grundlehren Math. Wiss. 230, Springer-Verlag, Berlin 1978. [86℄ O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural' eva, Linear and Quasilinear Equations of Paraboli Type, Amer. Math. So . Providen e, RI, 1968. [87℄ J. Langer and D.A. Singer, Curve Straightening and a minimax argument for losed elasti urves, Topology 24 (1985), 241{247. [88℄ G.M. Lieberman, Se ond Order Paraboli Dierential Equations, World S ienti , Singapore 1996.

© 2001 by Chapman & Hall/CRC

252

Bibliography

[89℄ E. Meron, Pattern formation in ex itable media, Phys. Rep. 218 (1992), 1{66. [90℄ J.D. Murray, Mathemati al Biology, Springer-Verlag, NY, 1989. [91℄ K. Nakayama, H. Segur and M. Wadati, Integrability and the motion of

urves, Phys. Rev. Lett. 69, 2603{2606. [92℄ J.A. Oaks, Singularities and self-interse tions of urves evolving on surfa es, Indiana Univ. Math. J. 43 (1994), 959{981. [93℄ P. J. Olver, Equivalen e, Invariants and Symmetry, Cambridge Univ. Press, Cambridge, 1995. [94℄ P.J. Olver, G. Sapiro and A. Tannenbaum, Dierential invariant signatures and ows in omputer vision: A symmetry group approa h, GeometryDriven Diusion in Computer Vision, B.M. Ter Haar Romeny, Ed., 205{306, Kluwer, Dordre ht, 1994. [95℄ A. Polden, Evolving urves, Honours thesis, Australian National Univ. 1991. [96℄ M.H. Protter and H.F. Weinberger, Maximum Prin iples in Dierential Equations, Prenti e-Hall, 1967. [97℄ G. Sapiro and A. Tannenbaum, On invariant urve evolution and image analysis, Indiana Univ. J. Math. 42 (1993), 985{1009. [98℄ G. Sapiro and A. Tannenbaum, On aÆne plane urve evolution, J. Fun t. Anal. 119 (1994), 79{120. [99℄ R. S hneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993. [100℄ H.M. Soner, Motion of a set by the urvature of its boundary, J. Di. Eqn's 101 (1993), 313{372. [101℄ P.E. Souganidis, Front propagation: Theory and appli ations, 186{242, Le ture Notes in Math. 1660, Springer-Verlag, Berlin, 1997. [102℄ A. Stahl, Convergen e of solution to the mean urvature ows with a Neumann boundary ondition, Cal . Var PDEs 4 (1996), 421{441. [103℄ A Stan u, Uniqueness of self-similar solutions for a rystalline ow, Indiana Univ. Math. J. 45 (1996), 1157{1174. [104℄ A. Stan u, Asymptoti behaviour of solutions to a rystalline ow, Hokkaido Math. J. 27 (1998), 303{320. [105℄ M. Struwe, On the evolution of harmoni maps in higher dimensions, J. Dierential Geom. 28 (1988), 485{502. [106℄ J.E. Taylor, Constru tions and onje tures in rystalline nondierential geometry, Dierential Geometry, 321{336, Pitman Mono. Surveys Pure Appl. Math., 52, Longman, Harlow, 1991. [107℄ J.E. Taylor, Motion of urves by rystalline urvature, in luding triple jun tions and boundary points, Di. Geom.: Partial Di. Eqn's on Manifold (Los Angeles, 1990), 417{438, Pro . Sympos. Pure Math. 54, Part 1, Amer. Math. So ., Providen e, RI, 1993.

© 2001 by Chapman & Hall/CRC

Bibliography

253

[108℄ J.I.E. Urbas, On the expansion of starshaped hypersurfa es by symmetri fun tion of their prin ipal urvatures, Math. Z. 205 (1990), 355{372. [109℄ J.I.E. Urbas, Convex urves moving homotheti ally by negative powers of their urvature, Asian J. Math. 3 (1999), 635{658. [110℄ Y. Wen, L2 - ow of urve straightening in the plane, Duke Math. J. 70 (1993), 683{698. [111℄ Y. Wen, Curve straightening ow deforms losed plane urves with nonzero rotation number to ir les, J. Dierential Eqn's. 120 (1995), 89{107. [112℄ E. Zeilder, Nonlinear Fun tional Analysis and its Appli ations, Vol. 1, Springer-Verlag, NY, 1985. [113℄ X.P. Zhu, Aysmptoti behaviour of anisotropi urve ows, J. Dierential Geom. 48 (1998), 225{274.

© 2001 by Chapman & Hall/CRC