Gauss Curvature Flow Self-similar Solutions And Spherical

In the thesis we mainly study when a closed convex surface satisfying K=<x,v> is the unit sphere? Here K is Gauss curvature, x is the position vector, v is the unit outer normal vector of the surface.In the introduction, we introduce the history from curve-shortening flow to mean curvature flow to Gauss curvature flow and the background knowledge, as well as self-similar solutions to the Gauss curvature flow. At the same time, we present the main result of this thesis.In Chapter 1,we review the involved basic concepts of Riemannian geometry and some related knowledge about the curvature and the equations of Gauss and Codazzi.In Chapter 2,we do some related calculation about two-dimensional Gauss curva-ture flow, which is beneficial to our calculation of the elliptical situation.In Chapter 3,we mainly verify that the two-dimension closed convex surface which satisfies the equation above must be a unit sphere in the three-dimensional Euclidean space.In Chapter 4, using maximum principle we discuss that when a closed convex surface of three dimensions which satisfies the equation above is the unit sphere in four Euclidean space.