The curve shortening problem

Kai-Seng Chou, Xi-Ping Zhu1584882131, 9781584882138, 9781420035704

Although research in curve shortening flow has been very active for nearly 20 years, the results of those efforts have remained scattered throughout the literature. For the first time, The Curve Shortening Problem collects and illuminates those results in a comprehensive, rigorous, and self-contained account of the fundamental results.The authors present a complete treatment of the Gage-Hamilton theorem, a clear, detailed exposition of Grayson’s convexity theorem, a systematic discussion of invariant solutions, applications to the existence of simple closed geodesics on a surface, and a new, almost convexity theorem for the generalized curve shortening problem.Many questions regarding curve shortening remain outstanding. With its careful exposition and complete guide to the literature, The Curve Shortening Problem provides not only an outstanding starting point for graduate students and new investigations, but a superb reference that presents intriguing new results for those already active in the field.

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Table of contents :
The Curve Shortening Problem……Page 2
CONTENTS……Page 4
PREFACE……Page 6
1.1 Short time existence……Page 9
1.2 Facts from the parabolic theory……Page 23
1.3 The evolution of geometric quantities……Page 27
Notes……Page 30
Bibliography……Page 250
2.1 Travelling waves……Page 35
2.2 Spirals……Page 37
2.3 The support function of a convex curve……Page 41
2.4 Self-similar solutions……Page 43
Notes……Page 50
3.1 Blaschke Selection Theorem……Page 52
3.2 Preserving convexity and shrinking to a point……Page 54
3.3 Gage-Hamilton Theorem……Page 58
3.4 The contracting case of the ACEF……Page 66
3.5 The stationary case of the ACEF……Page 80
3.6 The expanding case of the ACEF……Page 87
Notes……Page 93
Chapter 4: The Convex Generalized Curve Shortening Flow……Page 99
4.1 Results from the Brunn-Minkowski Theory……Page 100
4.2 The AGCSF for sigma in (1/3,1)……Page 103
4.3 The affne curve shortening flow……Page 108
4.4 Uniqueness of self-similar solutions……Page 118
Notes……Page 121
5.1 An isoperimetric ratio……Page 126
5.2 Limits of the rescaled
ow……Page 134
5.3 Classification of singularities……Page 139
Notes……Page 144
Chapter 6: A Class of Non-convex Anisotropic Flows……Page 148
6.1 The decrease in total absolute curvature……Page 149
6.2 The existence of a limit curve……Page 152
6.3 Shrinking to a point……Page 158
6.4 A whisker lemma……Page 165
6.5 The convexity theorem……Page 169
Notes……Page 181
Chapter 7: Embedded Closed Geodesics on Surfaces……Page 183
7.1 Basic results……Page 184
7.2 The limit curve……Page 190
7.3 Shrinking to a point……Page 192
7.4 Convergence to a geodesic……Page 200
Notes……Page 206
Chapter 8: The Non-convex Generalized Curve Shortening Flow……Page 207
8.1 Short time existence……Page 208
8.2 The number of convex arcs……Page 215
8.3 The limit curve……Page 222
8.4 Removal of interior singularities……Page 232
8.5 The almost convexity theorem……Page 243
Notes……Page 249