On The Embedding Research For A Plane Curve Flow

As one of the main topics of modern mathematical research, "curve 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.Let F(u, t) : S~1×[0, T)→R~2 be a family of closed plane curves, thendefines the curve shrinking flow, where k = k(u, t) is the signed curvature of the evolving curve, and N = N(u,t) the unit inward pointing normal vector along the curve. In 1986, Gage-Hamilton proved that any convex curve preserves convexity, and will be more and more circular under the curve shrinking flow. And furthermore, it will converge to a round point in finite time. In 1987, Grayson proved that any embedded curve will become convex in finite time under the curve shrinking flow. As a result, he got that any embedded curve converges to a round point in finite time by Gage-Hamilton theorem. Since it needs very strong skills to prove Grayson's theorem, Hamilton and Huisken introduced their new proofs of Grayson's theorem respectively in 1992 and 1999.There are two aims in this paper. First, we introduce the methods of Hamilton and Huisken who used three isoperimetric ratios and proved that the three isoperimetric ratios increase under the curve shrinking flow. As a result, they got their new proofs of Grayson's theorem. The new proofs of Hamilton and Huisken for Grayson's theorem all need the condition that embedded curves stay embedded under the curve shrinking flow. As a result, it is necessary to research the embedding of curve flows. The other aim of this paper is to research the embedding of a curve flow. Let F(u, t) : S~1×[0, T)→R~2 be a smooth solution of the flowwhere k = k(u, t) is the signed curvature of the evolving curve, and N = N(u, t) the unit inward pointing normal vector along the curve. And if the first curve F(u, 0) is embedded, F(u, t) stay embedded for any t.