Mean curvature flow with surgeries of two–convex hypersurfaces

Abstract

We consider a closed smooth hypersurface immersed in euclidean space evolving by mean curvature flow. It is well known that the solution exists up to a finite singular time at which the curvature becomes unbounded. The purpose of this paper is to define a flow after singularities by a new approach based on a surgery procedure. Compared with the notions of weak solutions existing in the literature, the flow with surgeries has the advantage that it keeps track of the changes of topology of the evolving surface and thus can be applied to classify all geometries that are possible for the initial manifold. Our construction is inspired by the procedure originally introduced by Hamilton for the Ricci flow, and then employed by Perelman in the proof of Thurston's geometrization conjecture. In this paper we consider initial hypersurfaces which have dimension at least three and are two-convex, that is, such that the sum of the two smallest principal curvatures is nonnegative everywhere. Under these assumptions, we construct a flow with surgeries which has uniformly bounded curvature until the evolving manifold is split in finitely many components with known topology. As a corollary, we obtain a classification up to diffeomorphism of the hypersurfaces under consideration.