The Curve Shortening Problem

The curve shortening flow and other related geometric evolution equations serve as mathematical models for applications in diverse areas, such as phase transitions, flame front propagation, chemical reaction, mathematical biology, and image processing. This book offers a comprehensive account of the fundamental results relevant to these flows.

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.