A Class of Non-convex Anisotropic Flows
Kai-Seng Chou, Xi-Ping Zhu · 2001
A Class of Non-convex Anisotropic Flows In this chapter, we continue the study of ows for non-convex curves. Let and be two smooth, 2-period functions of the tangent angle satisfying () > 0 (6.1) and ( + ) = () ; ( + ) = () : (6.2) We consider the Cauchy problem for @ @t = (k + )n ; (6.3) where the initial curve is a smooth, embedded closed curve. This ow may be regarded as the linear case for the general ow (1.2), where F is uniformly parabolic and symmetric. Remember that the condition (6.2) means that F is symmetric. Without this condition, embeddedness may not be preserved under the ow. When is convex, we have shown in x3.2 that the ow also preserves convexity and it shrinks to a point where ! is nite. In this chapter we shall show that the Grayson convexity theorem holds for (6.3).