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

Abstract

This paper analyzes the singular behavior of the mean curvature flow generated by the boundary of the compact mean-convex region of R n + 1 \mathbf {R}^{n+1} or of an ( n + 1 ) (n+1) -dimensional riemannian manifold. If n > 7 n>7 , 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 n ≥ 7 n\ge 7 , the same results are shown up to the first time that singularities occur.