The nature of singularities in mean curvature flow of mean-convex sets

This paper analyzes the singular behavior of the mean curvature flow generated by the boundary of the compact mean-convex region of or of an -dimensional riemannian manifold. If , the moving boundary is shown to be very nearly convex in a spacetime neighborhood of any singularity. In particular, the tangent flows at singular points are all shrinking spheres or shrinking cylinders. If , the same results are shown up to the first time that singularities occur.