Blow-up Continuity for Type-I, Mean-Convex Mean Curvature Flow

TL;DR

This paper proves the continuity of the first singular time and limit set in certain mean-convex mean curvature flows, using neck-pinching arguments and geometric conditions to understand singularity formation.

Contribution

It establishes the continuity of singular times and limit sets for mean-convex flows, combining neck-pinching techniques with non-collapsing conditions and tangent flow uniqueness.

Findings

Continuity of first singular time under mean curvature flow.

Use of neck-pinching argument to analyze singularities.

Application of non-collapsing condition and tangent flow uniqueness.

Abstract

Under mean curvature flow, a closed, embedded hypersurface $M(t)$ becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time $T$ and the limit set "$M(T)$", with respect to initial data. We employ an Angenent-like neck-pinching argument to force singularities in nearby flows. However, since we cannot prescribe initial data, we combine Andrews' $\alpha$-non-collapsed condition and Colding and Minicozzi's uniqueness of tangent flows to place appropriately sized spheres in the region inside the hypersurface.