The Curve Shortening Problem

Kai-Seng Chou, Xi-Ping Zhu9781584882138, 1-58488-213-1

The curve shortening flow along with other closely related geometric evolution equations serve as mathematical models for applications in diverse areas, such as phase transition, flame front propagation, chemical reaction, mathematical biology, and image processing. The first book dedicated to this subject, The Curve Shortening Problem presents a rigorous, comprehensive account of the fundamental results relevant to these flows.

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Table of contents :
CONTENTS……Page 6
PREFACE……Page 8
1.1 Short time existence……Page 12
1.2 Facts from the parabolic theory……Page 26
1.3 The evolution of geometric quantities……Page 30
2.1 Travelling waves……Page 38
2.2 Spirals……Page 40
2.3 The support function of a convex curve……Page 44
2.4 Self-similar solutions……Page 46
3.1 Blaschke Selection Theorem……Page 56
3.2 Preserving convexity and shrinking to a point……Page 58
3.3 Gage-Hamilton Theorem……Page 62
3.4 The contracting case of the ACEF……Page 70
3.5 The stationary case of the ACEF……Page 84
3.6 The expanding case of the ACEF……Page 91
4 The Convex Generalized Curve Shortening Flow……Page 104
4.1 Results from the Brunn-Minkowski The- ory……Page 105
4.2 The AGCSF for……Page 108
4.3 The aÆne curve shortening
ow……Page 113
4.4 Uniqueness of self-similar solutions……Page 123
5.1 An isoperimetric ratio……Page 132
5.2 Limits of the rescaled
ow……Page 140
5.3 Classication of singularities……Page 145
6 A Class of Non-convex Anisotropic Flows……Page 154
6.1 The decrease in total absolute curvature……Page 155
6.2 The existence of a limit curve……Page 158
6.3 Shrinking to a point……Page 164
6.4 A whisker lemma……Page 171
6.5 The convexity theorem……Page 175
7 Embedded Closed Geodesics on Surfaces……Page 190
7.1 Basic results……Page 191
7.2 The limit curve……Page 197
7.3 Shrinking to a point……Page 199
7.4 Convergence to a geodesic……Page 207
8 The Non-convex Generalized Curve Shortening Flow……Page 214
8.1 Short time existence……Page 215
8.2 The number of convex arcs……Page 222
8.3 The limit curve……Page 229
8.4 Removal of interior singularities……Page 239
8.5 The almost convexity theorem……Page 250
Bibliography……Page 258
Index……Page 265