Some Notes On Plane Embedded Curve Shortening Flow

As one of the main topics of modern mathematical research, "geometry flow" is to study the deformation of geometric object with its applications by the tools of geometry and analysis method. Since the 1980s, it has always been one of the hot research in the field of geometric analysis, we try to study some relate problems on the background of above researches.In this dissertation, we mainly discusses the curve shortening flow. Given difficulty of the question and the current level of author, we only consider the plane curve under the mean curvature way. For this particular mean curvature flow, we usually call it "curve shortening flow", referred to as curve flow.We consider the following planar curve shortening flow:where γ₀ is a simple embedded plane curve is the curvature of γ(·, t), N is the unit normal vector of γ(·,t). We will consider its solution in the local existence, the plane curve keeping embedded under the curve evolution and the nature of long time behavior. Finally, the greatest limited time T of the curve shortening flow is estimated.This dissertation consists of three sections. In the first one, we wili briefly talk about the history and background of curvature flow theory. In section two, we will give some of the results about planar curve shortening flow. Section three will give the main results of this dissertation.