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 (i) for $n\leq 7$, and (ii) for arbitrary $n$ up to the first singular time without the smooth condition for $M_{\infty}$.