Subsequent singularities of mean convex mean curvature flows in smooth manifolds

Abstract

For any n-dimensional smooth manifold \(\Sigma \), we show that all the singularities of the mean curvature flow with any initial mean convex hypersurface in \(\Sigma \) are cylindrical (of convex type) if the flow converges to a smooth hypersurface \(M_\infty \) (maybe empty) at infinity. Previously this was shown (1) for \(n\le 7\), and (2) for arbitrary n up to the first singular time without the smooth condition on \(M_\infty \).