Asymptotic convergence for a class of anisotropic curvature flows

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