Mean convex flows with surgery

Abstract

We construct a mean curvature flow with surgery starting with any compact mean convex hypersurface in $\mathbb{R}^{n+1}$, extending previous results of Huisken-Sinestrari, Brendle-Huisken, and Haslhofer-Kleiner for 2-convex flows. In contrast with previous constructions of mean curvature flow with surgery, the topological surgeries are performed by the flow itself through nondegenerate cylindrical singularities. At the same time the flow needs to be slightly adjusted at finitely many smooth times.