A Class Of Inverse Curvature Flows In Rⁿ⁺¹

The study of inverse curvature flows is one of the hot topics in Geometric Analysis.Inverse curvature flows can be used to derive some important geometric inequalities,and so they attract the attention of geometers from domestic and abroad.We consider the evolution of closed,star-shaped,admissible hypersurfaces in Rn+1 along the inverse curvature flow X=|X|α1Fβ.α≤1,β>0,where,F is a nonnegative symmetric function of principal curvatures,homogeneous of degree one,and satisfies certain monotonicity and concavity.v is the unit outward normal vec-tor of the evolving hypersurface Mt:=X(Sn,t),and |X| is the distance from the point X(x,t)to the origin of Rn+1.We prove that for the case α≤1,β>0,α+β≤2,this flow exists for all the time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling.Besides,for the case α≤1,α+β>2,if furthermore the initial closed hypersurface is strictly convex,then the strict convexity is preserved during the evolution process and the flow blows up at finite time.This conclusion ex-tends the results of Professor Claus Gerhardt,Professor John I.E.Urbas,Professor J.Mao et al.on the theory of inverse curvature flows.This thesis is based on the prvious joint-work[13]with my collaborators.The main body of this thesis consists of three parts.More precisely,they are:In Chapter 1,we give an introduction to the research background,the current research situation of inverse curvature flows,and also the main conclusions of this thsis.In Chapter 2,we give some formulae for star-shaped hypersurfaces in Rn+1,and then use these formulae to get the scalar version of the inverse curvature flow consid-ered.In Chapter 3,C0-estimate,the gradient estimate and C2-estimate will be given for the solution of the scalar flow equation,and together with the krylov-Safonov estimate method for the second-order parabolic partial differential equations we will give the long-time existence of the inverse curvature flow,and also the accurate description of the asymptotic behavior of the flow.Besides,we can also analyze and explain the blow up of the flow at finite time when α,β choose values in different ranges.