Diameter and curvature control under mean curvature flow

TL;DR

This paper establishes uniform diameter bounds and sharp regularity estimates for two-convex hypersurfaces evolving under mean curvature flow, providing new insights into singularity formation and geometric control.

Contribution

It proves diameter control near singularities and derives sharp regularity estimates for level set flow of two-convex hypersurfaces, advancing understanding of mean curvature flow behavior.

Findings

Intrinsic diameter remains bounded near singularities.

Sharp $L^{n-1}$-estimates for regularity scale.

Results apply even to classical mean convex surfaces in $\

Abstract

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^{n-1}$-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis of cylindrical regions ($\varepsilon$-tubes) under mean curvature flow. The results are new even in the most classical case of mean convex surfaces evolving by mean curvature flow in $\mathbb{R}^3$.