Subsequent singularities of mean convex mean curvature flows in smooth manifolds

TL;DR

This paper proves that in smooth manifolds, mean convex mean curvature flows with certain convergence properties develop only cylindrical singularities, extending previous results to all dimensions under these conditions.

Contribution

It generalizes the classification of singularities in mean curvature flow to all dimensions when the flow converges smoothly at infinity.

Findings

All singularities are cylindrical if the flow converges smoothly at infinity.

Previous results were limited to dimensions up to 7 or without smooth convergence.

The work extends singularity classification to higher dimensions under new conditions.

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 $M_{\infty}$.