The curve shortening problem

256 Pages·2001·0.931 MB·English

by  Kai-Seng Chou, Xi-Ping Zhu

Curve
The
Shortening
Problem
© 2001 by Chapman & Hall/CRC
Curve
The
Shortening
Problem
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

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© 2001 by Chapman & Hall/CRC
v
CONTENTS
Preface vii
1 Basic Results 1
1.1 Short time existence . . . . . . . . . . . . . . . . . . . 1
1.2 Facts from the parabolic theory . . . . . . . . . . . . . 15
1.3 The evolution of geometric quantities . . . . . . . . . . 19
2 Invariant Solutions for the Curve Shortening Flow 27
2.1 Travelling waves . . . . . . . . . . . . . . . . . . . . . 27
2.2 Spirals . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 The support function of a convex curve . . . . . . . . 33
2.4 Self-similar solutions . . . . . . . . . . . . . . . . . . . 35
3 The Curvature-Eikonal Flow for Convex Curves 45
3.1 Blaschke Selection Theorem . . . . . . . . . . . . . . . 45
3.2 Preserving convexity and shrinking to a point . . . . . 47
3.3 Gage-Hamilton Theorem . . . . . . . . . . . . . . . . . 51
3.4 The contracting case of the ACEF . . . . . . . . . . . 59
3.5 The stationary case of the ACEF . . . . . . . . . . . . 73
3.6 The expanding case of the ACEF . . . . . . . . . . . . 80
4 The Convex Generalized Curve Shortening Flow 93
4.1 Results from the Brunn-Minkowski Theory . . . . . . 94
4.2 The AGCSF for (cid:27) in (1/3, 1) . . . . . . . . . . . . . . 97
4.3 The aÆne curve shortening (cid:13)ow . . . . . . . . . . . . . 102
4.4 Uniqueness of self-similar solutions . . . . . . . . . . . 112
5 The Non-convex Curve Shortening Flow 121
5.1 An isoperimetric ratio . . . . . . . . . . . . . . . . . . 121
5.2 Limits of the rescaled (cid:13)ow . . . . . . . . . . . . . . . . 129
© 2001 by Chapman & Hall/CRC
vi
5.3 Classi(cid:12)cation of singularities. . . . . . . . . . . . . . . 134
6 A Class of Non-convex Anisotropic Flows 143
6.1 The decrease in total absolute curvature . . . . . . . . 144
6.2 The existence of a limit curve . . . . . . . . . . . . . . 147
6.3 Shrinking to a point . . . . . . . . . . . . . . . . . . . 153
6.4 A whisker lemma . . . . . . . . . . . . . . . . . . . . . 160
6.5 The convexity theorem . . . . . . . . . . . . . . . . . . 164
7 Embedded Closed Geodesics on Surfaces 179
7.1 Basic results. . . . . . . . . . . . . . . . . . . . . . . . 180
7.2 The limit curve . . . . . . . . . . . . . . . . . . . . . . 186
7.3 Shrinking to a point . . . . . . . . . . . . . . . . . . . 188
7.4 Convergence to a geodesic . . . . . . . . . . . . . . . . 196
8 The Non-convex Generalized Curve Shortening
Flow 203
8.1 Short time existence . . . . . . . . . . . . . . . . . . . 204
8.2 The number of convex arcs . . . . . . . . . . . . . . . 211
8.3 The limit curve . . . . . . . . . . . . . . . . . . . . . . 218
8.4 Removal of interior singularities . . . . . . . . . . . . . 228
8.5 The almost convexity theorem. . . . . . . . . . . . . . 239
Bibliography 247
© 2001 by Chapman & Hall/CRC
vii
PREFACE
A geometric evolution equation (for plane curves) is of the form
@(cid:13)
=fn ; ((cid:3))
@t
where (cid:13)((cid:1);t) is a family of curves with a choice of continuous unit
normal vector n((cid:1);t) and f is a function dependingon the curvature
of(cid:13)((cid:1);t)withrespectton((cid:1);t). Anysolutionof((cid:3)) isinvariantunder
the Euclidean motion. The simplest geometric evolution equation is
the eikonal equation when f is taken to be a non-zero constant.
The next one is the curvature-eikonal (cid:13)ow when f is linear in the
curvature. It includes the curve shortening (cid:13)ow (CSF)
@(cid:13)
=kn ;
@t
as a special case. Let L(t) be the perimeter of a family of closed
curves (cid:13)((cid:1);t) driven by ((cid:3)). We have the (cid:12)rst variation formula
dL
(t) = (cid:0) fkds :
dt Z(cid:13)((cid:1);t)
2
Therefore, the CSF is the negative L -gradient (cid:13)ow of the length.
When (cid:13)((cid:1);t) is also embedded, its enclosed area satis(cid:12)es
dA
(t) =(cid:0)2(cid:25) :
dt
Thus, any embedded closed curve shrinks under the (cid:13)ow and ceases
to exist beyond A(0)=2(cid:25). The following two results completely char-
acterize the motion.
Theorem A (Gage-Hamilton) The CSF preserves convexity and
shrinksany closed convex curve to a point. Furthermore, ifwe dilate
the (cid:13)owsothat itsenclosedarea isalways equalto (cid:25), thenormalized
(cid:13)ow converges to a unit circle.
Theorem B (Grayson) The CSF starting at any closed embedded
curve becomes convex at some time before A(0)=2(cid:25).
© 2001 by Chapman & Hall/CRC
viii
From the analytic point of view, the curvature of (cid:13)((cid:1);t) satis(cid:12)es
3
kt =kss+k ;
where s = s(t) is the arc-length parameter of (cid:13)((cid:1);t). This is a non-
linear heat equation with superlinear growth. It is clear that the
curvature must blow up in (cid:12)nite time. However, it is the geometric
natureofthe(cid:13)ow thatenablesonetoobtainpreciseresultslikethese
two theorems. On the other hand, the CSF is a special case of the
mean curvature (cid:13)ow for hypersurfaces. It turns out that, although
Theorem A continues to hold for the mean curvature (cid:13)ow, Theorem
B does not. This makes planar (cid:13)ows special among curvature (cid:13)ows.
After Theorems A and B, subsequent works on the CSF go in
two directions. One is to study the structure of the singularities
of the (cid:13)ow for immersed curves, and the other is to consider more
general planar (cid:13)ows. In this book, we present a complete treatment
on Theorem A and Theorem B as well as provide some of general-
izations. There are eight chapters. We outline the content of each
chapter as follows: In Chapter 1, we discuss basic results such as
local existence, separation principle, and (cid:12)niteness of nodes for the
general (cid:13)ow ((cid:3)) under the parabolic assumption. In Chapter 2, we
describe special solutions of the CSF which arise from its Euclidean
andscalinginvariance: travellingwaves, spirals,andcontracting and
expanding self-similar solutions. These solutions will become im-
portant in the classi(cid:12)cation of singularities for the CSF. Theorem
A is proved in Chapter 3. In the same chapter, we also study the
anisotropic curvature-eikonal (cid:13)ow
@(cid:13)
=((cid:8)(n)k+(cid:9)(n)) ; (cid:8)>0 :
@t
This (cid:13)ow may be viewed as the CSF in a Minkowski geometry when
(cid:9) (cid:17) 0 and its general form is proposed as a model in phase transi-
tion. Dependingontheinhomogeneousterm(cid:9), itshrinksto apoint,
expands to in(cid:12)nity, or converges to a stationary solution. We deter-
mine its asymptotic behaviour in all these cases. In Chapter 4, we
study the anisotropic generalized CSF
@(cid:13) =(cid:8)(n)jkj(cid:27)(cid:0)1kn ; (cid:8)>0 ; (cid:27) >0 ;
@t
following the work of Andrews [8] and [10]. Analogues of Theorem
A are proved for (cid:27) 2 [1=3;1). When (cid:27) = 1=3 and (cid:8) (cid:17) 1, the (cid:13)ow is
© 2001 by Chapman & Hall/CRC
ix
aÆne invariant and is proposed in connection with image processing
and computer vision. Beginning from Chapter 5, we turn to non-
convex curves. First, we present a relatively short proof of Theorem
B which is based on the blow-up and the classi(cid:12)cation of singulari-
ties. This approach has been successfully adopted in many geomet-
ricproblemsincludingnonlinearheatequations,harmonicheat(cid:13)ows,
Ricci(cid:13)ows,andthemeancurvature(cid:13)ow. Next,wepresentGrayson’s
geometric approach where the Sturm oscillation theorem is used in
an essential way in Chapter 6. Though strictly two-dimensional, it
is powerful and works for a large class of uniformly parabolic (cid:13)ows
((cid:3)). In Chapter 7, we discuss how the CSF can be used to prove
the existence of embedded, closed geodesics on a surface. Finally, in
Chapter 8, we study the isotropic generalized CSF and establish an
almost convexity theorem when (cid:27) 2 (0;1). Whether the convexity
theorem holds for this class of (cid:13)ows remains an unsolved problem.
Many interesting results on ((cid:3)) have been obtained in the past
(cid:12)fteen years. It is impossibleto include all of them in a book of this
size. Apart from a thorough discussion on Theorem A and B, the
choice of the rest of the material in this book is rather subjective.
Some are based on our work on this topic. To balance things the
we sketch the physical background, describe related results, and oc-
casionally point out some unsolved problems in the notes which can
be found at the end of each chapter. We hope that the reader can
gain a panoramic view through them. We shallnot discussthe level-
set approach to curvature (cid:13)ows in spite of its popularity. Here we
are mainlyconcerned withsingularitiesand asymptotic behaviour of
planar (cid:13)ows where the classical approach is suÆcient.
Thanks are due to Dr. Sunil Nair for proposing the project,
and to Ms.Judith Kamin for her e(cid:11)ort in editing the book. We are
also indebted to the Earmarked Grant of Research, Hong Kong, the
FoundationofOutstandingYoungScholars,andtheNationalScience
FoundationofChinafortheirsupportinourworkoncurvature(cid:13)ows,
some of which has been incorporated in this book.
© 2001 by Chapman & Hall/CRC
Chapter 1
Basic Results
Inthischapter, we (cid:12)rst establishthe existence of a maximalsolution
and some basic qualitative behaviour such as the separation princi-
ple and (cid:12)niteness of nodes for the general (cid:13)ow (1.2). These proper-
ties are direct consequences of the parabolic nature of the (cid:13)ow. For
the reader’sconvenience, we collect fundamentalresultsonparabolic
equations in Section 2. In particular, the \Sturm oscillation theo-
rem," which is not found in standard texts on this subject, will play
an important role in the removal of singularities of the (cid:13)ow. In Sec-
tion 3, wederiveevolutionequationsforvariousgeometric quantities
ofthe(cid:13)ow. Theywillbecomeimportantwhenwestudythelongtime
behaviour of the (cid:13)ow.
1.1 Short time existence
1
We begin by recalling the de(cid:12)nition of a curve. An immersed, C -
curve is a continuously di(cid:11)erentiable map (cid:13) from I to R2 with a
non-zero tangent (cid:13)p = d(cid:13)=dp. Throughout this book, I is either an
1
interval or an arc of the unit circle S , and a curve always means an
1
immersed, C -curve unless speci(cid:12)ed otherwise. The curve is closed
1
© 2001 by Chapman & Hall/CRC
2 CH. 1. Basic Results
if I is the unit circle. It is embedded if it is one-to-one. Given a
1 2
curve (cid:13) = ((cid:13) ;(cid:13) ), its unit tangent is given by t = (cid:13)p=(cid:13)p and its
j j
2 1
unit normal, n, is given by ( (cid:13)p;(cid:13)p)= (cid:13)p . When (cid:13) is an embedded
(cid:0) j j
closed curve and t runs in the counterclockwise direction, n is the
inner unit normal. The tangent angle of the curve is the angle (cid:18)
between the unit tangent and the positive x-axis. It is de(cid:12)ned as
modulo 2(cid:25). However, once the tangent angle at a certain point on
the curveisspeci(cid:12)ed,achoiceofcontinuoustangent angles alongthe
(cid:13)ow is determined uniquely. The curvature of (cid:13) with respect to n,
k, is de(cid:12)ned via the Frenet formulas,
dt dn
=kn; = kt; (1.1)
ds ds (cid:0)
where ds= (cid:13)p dp is the arc-length element. Explicitly we have
j j
2 1 1 2
(cid:13)pp(cid:13)p (cid:13)pp(cid:13)p
k = (cid:0) :
3
(cid:13)p
j j
We shall study the (cid:13)ow
@(cid:13)
=F((cid:13);(cid:18);k)n; (p;t) I (0;T); T >0; (1.2)
@t 2 (cid:2)
where F =F(x;y;(cid:18);q) isa given functioninR2 R R, 2(cid:25)-periodic
(cid:2) (cid:2)
in (cid:18). A (classical) solution to (1.2) is a map (cid:13) from I (0;T)
(cid:2)
to R2 satisfying (i) it is continuously di(cid:11)erentiable in t and twice
continuously di(cid:11)erentiable in p, (ii) for each t, p (cid:13)(p;t) is a
7(cid:0)!
curve, and (iii) (cid:13) satis(cid:12)es (1.2) where n and k are respectively the
unitnormaland curvatureof (cid:13)(;t) withrespect to n. Given a curve
(cid:1)
(cid:13)0, we are mainly concerned with the following Cauchy problem:
To (cid:12)nd a solution of (1.2) which approaches (cid:13)0 as t # 0.
© 2001 by Chapman & Hall/CRC