The Curve Shortening Problem

ISBN-13: 9781584882138 / Angielski / Twarda / 2001 / 272 str.

Kai Seng Chou; XI-Ping Zhu

Opis

Although research in curve shortening flow has been very active for ne arly 20 years, the results of those efforts have remained scattered th roughout the literature. For the first time, The Curve Shortening Prob lem collects and illuminates those results in a comprehensive, rigorou s, and self-contained account of the fundamental results. The authors present a complete treatment of the Gage-Hamilton theorem, a clear, de tailed exposition of Grayson's convexity theorem, a systematic discuss ion of invariant solutions, applications to the existence of simple cl osed geodesics on a surface, and a new, almost convexity theorem for t he generalized curve shortening problem. Many questions regarding curv e shortening remain outstanding. With its careful exposition and compl ete guide to the literature, The Curve Shortening Problem provides not only an outstanding starting point for graduate students and new inve stigations, but a superb reference that presents intriguing new result s for those already active in the field.

Spis treści

Basic Results. Invariant Solutions for the Curve Shortening Flow. The Curvature-Eikonal Flow for Convex Curves. The Convex Generalized Curve Shortening Flow. The Non-Convex Curve Shortening Flow. A Class of Non-Convex Anisotropic Flows. Embedded Closed Geodesic on Surfaces. The Non-Convex Generalized Curve Shortening Flow. Bibliography.

O autorze

Chou, Kai-Seng; Zhu, Xi-Ping