Asymptotic convergence for a class of anisotropic curvature flows

Abstract

In this paper, by using new auxiliary functions, we study a class of contracting flows of closed, star-shaped hypersurfaces in Rn+1 with speed rαβσk1β, where σk is the k-th elementary symmetric polynomial of the principal curvatures, α, β are positive constants and r is the distance from points on the hypersurface to the origin. We obtain convergence results under some assumptions of k, α, β. When k≥2, 0<β≤1, α≥β+k, we prove that the k-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Li-Sheng-Wang's result from uniformly convex to k-convex. When k≥2, β=k, α≥2k, we prove that the k-convex solution to the flow exists for all time and converges smoothly to a sphere after normalization, in particular, we generalize Ling Xiao's result from k=2 to k≥2.