Convexity estimates for a nonhomogeneous mean curvature flow

We study the evolution of a closed immersed hypersurface whose speed is given by a function phi(H) of the mean curvature asymptotic to H/1n H for large H. Compared with other nonlinear functions of the curvatures, this speed has some good properties which allow for an easier study of the formation of singularities in the nonconvex case. We prove apriori estimates showing that any surface with positive mean curvature at the initial time becomes asymptotically convex near a singularity. Similar estimates also hold for the mean curvature flow; for the flow considered here they admit a simpler proof based only on the maximum principle.