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...

Mean curvature flow is a term that is used to describe the evolution of a hypersurface whose normal velocity is given by the mean curvature. In the simplest case of a convex closed curve on the plane, the properties of the mean curvature flow are described by Gage-Hamilton's theorem. This theorem states that under the mean curvature flow, the curve collapses to a point, and if the flow is diluted so that the enclosed area equals $pi$, the curve tends to the unit circle. In this book, the author gives a comprehensive account of fundamental results on singularities and the asymptotic behavior...