The Curve Shortening Problem

The Curve Shortening Problem

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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 more

Product details

  • Hardback | 272 pages
  • 160 x 238 x 22mm | 559.99g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 10 black & white illustrations
  • 1584882131
  • 9781584882138
  • 1,449,545

Table of contents

BASIC RESULTS Short Time Existence Facts from Parabolic Theory Evolution of Geometric Quantities INVARIANT SOLUTIONS FOR THE CURVE SHORTENING FLOW Travelling Waves Spirals The Support Function of a Convex Curve Self-Similar Solutions THE CURVATURE-EIKONAL FLOW FOR CONVEX CURVES Blaschke Selection Theorem Preserving Convexity and Shrinking to a Point Gage-Hamilton Theorem The Contracting Case of the ACEF The Stationary case of the ACEF The Expanding Case of the ACEF THE CONVEX GENERALIZED CURVE SHORTENING FLOW Results from Brunn-Minkowski Theory The AGCSF for s in (1/3,1) The Affine Curve Shortening Flow Uniqueness of Self-Similar Solutions THE NON-CONVEX CURVE SHORTENING FLOW An Isoperimetric Ratio Limits of the Rescaled Flow Classification of Singularities A CLASS OF NON-CONVEX ANISOTROPIC FLOWS Decrease in Total Absolute Curvature Existence of a Limit Curve Shrinking to a Point A Whisker Lemma The Convexity Theorem EMBEDDED CLOSED GEODESICS ON SURFACES Basic Results The Limit Curve Shrinking to a Point Convergence to a Geodesic THE NON-CONVEX GENERALIZED CURVE SHORTENING FLOW Short Time Existence The Number of Convex Arcs The Limit Curve Removal of Interior Singularities The Almost Convexity Theorem BIBLIOGRAPHYshow more