Evolution Of Convex Hypersurfaces By Powers Of The Mean Curvature

In this dissertation, we study the evolution of a closed,convex hypersurfaces in Rn+1 in direction of its normal vector, where the speed equals a positive power k of the mean curvature. We show that the flow exists on a maximal, finite time interval, and that, approaching the final time, the surfaces contract to a point.The dissertation is based on the article of Felix Schulze, I make up the details of calculation. By the calculation, I understand the basic equation of curvature flow well; on the other side my knowledge in curvature flow is improved and hope that I can understand and solve the problems of these aspects.The dissertation includes 3 chapters. In chapter 1, we briefly recall the history of mean curvature flow. In chapter 2, we introduce the basic evolution equations of curvature flow. In chapter 3, we use the evolution equations of curvature flow to prove the principal results:Theorem 1. Let F0:Mn→Rn+1 be a smooth immersion, whereH(F0(Mn)) 0. Then there exists a unique, smooth solution to the initial value problem (*) on a maximal, finite time interval [0,T). For k≥1 we have the bound T≥C(k,n)-1(?).In the case thati) F0(Mn) is strictly convex for 0< k< 1 ii) F0(Mn) weakly convex for k≥1 then the surfaces F(Mn,t) are strictly convex for all t> 0 and they contract for t→T to a point in Rn+1.For the corollary of Theorem 1, we have the following result of mean curvature flow: Theorem 2. In Euclid Space, there is a evolution equation for closed,convex hy-persurfaces:F0(·) is convex,there exists only one smooth solution on a maximal, finite time interval [0,T). F(·,t)is convex,the hypersurfaces contracts to a point as t→T.