Asymptotic Convergence for a Class of Fully Nonlinear Curvature Flows

In this paper, we study a class of contracting flows of closed, convex hypersurfaces in the Euclidean space R^{n + 1} with speed r^ασ_k, where σ_k is the k-th elementary symmetric polynomial of the principal curvatures, α∈ R¹, and r is the distance from the hypersurface to the origin. If α≥ k+ 1 , we prove that the flow exists for all time, preserves the convexity and converges smoothly after normalisation to a sphere centred at the origin. If α< k+ 1 , a counterexample is given for the above convergence. In the case k= 1 and α≥ 2 , we also prove that the flow converges to a round point if the initial hypersurface is weakly mean-convex and star-shaped.