The Curve Shortening Problem - Kai-Seng Chou

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The Curve Shortening Problem

Name: The Curve Shortening Problem
Brand: CHAPMAN & HALL
Price: 279.4 AUD
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By: Kai-Seng Chou, Xi-Ping Zhu

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Hardcover | 6 March 2001 | Edition Number 1

At a Glance

Hardcover
272 Pages

Dimensions(cm)
16.5 x 24.1 x 2.1

Hardcover

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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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You Can Find This Book In

Non-Fiction Mathematics Geometry Algebraic Geometry Engineering & Technology Technology in General Engineering in General Algebra Differential & Riemannian Geometry Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics

Preface	p. vii
Basic Results	p. 1
Short time existence	p. 1
Facts from the parabolic theory	p. 15
The evolution of geometric quantities	p. 19
Invariant Solutions for the Curve Shortening Flow	p. 27
Travelling waves	p. 27
Spirals	p. 29
The support function of a convex curve	p. 33
Self-similar solutions	p. 35
The Curvature-Eikonal Flow for Convex Curves	p. 45
Blaschke Selection Theorem	p. 45
Preserving convexity and shrinking to a point	p. 47
Gage-Hamilton Theorem	p. 51
The contracting case of the ACEF	p. 59
The stationary case of the ACEF	p. 73
The expanding case of the ACEF	p. 80
The Convex Generalized Curve Shortening Flow	p. 93
Results from the Brunn-Minkowski Theory	p. 94
The AGCSF for [sigma] in (1/3, 1)	p. 97
The affine curve shortening flow	p. 102
Uniqueness of self-similar solutions	p. 112
The Non-convex Curve Shortening Flow	p. 121
An isoperimetric ratio	p. 121
Limits of the rescaled flow	p. 129
Classification of singularities	p. 134
A Class of Non-convex Anisotropic Flows	p. 143
The decrease in total absolute curvature	p. 144
The existence of a limit curve	p. 147
Shrinking to a point	p. 153
A whisker lemma	p. 160
The convexity theorem	p. 164
Embedded Closed Geodesics on Surfaces	p. 179
Basic results	p. 180
The limit curve	p. 186
Shrinking to a point	p. 188
Convergence to a geodesic	p. 196
The Non-convex Generalized Curve Shortening Flow	p. 203
Short time existence	p. 204
The number of convex arcs	p. 211
The limit curve	p. 218
Removal of interior singularities	p. 228
The almost convexity theorem	p. 239
Bibliography	p. 247
Index	p. 254
Table of Contents provided by Syndetics. All Rights Reserved.

The Curve Shortening Problem

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Hardcover

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