Asymptotic convergence for a class of contracting curvature flows

In this paper, we study a class of fully nonlinear contracting curvature flows of closed, uniformly convex hypersurfaces in the Euclidean space [Formula: see text] with the normal speed [Formula: see text] given by [Formula: see text] or [Formula: see text], where [Formula: see text] is a monotone, symmetric, inverse-concave, homogeneous of degree one function of the principal curvatures, [Formula: see text] is the distance from the hypersurface to the origin and [Formula: see text] is the support function of hypersurface. If [Formula: see text], we prove that the flow exists for all time, and converges after normalization exponentially in the [Formula: see text] topology to a sphere centered at the origin. If [Formula: see text], a counterexample is given for the above convergence when speed equals to [Formula: see text].