An $\varepsilon$-regularity Theorem For The Mean Curvature Flow

TL;DR

This paper establishes an $$-regularity theorem for mean curvature flow, showing small energy conditions prevent singularities and imply the singular set has zero 2D Hausdorff measure.

Contribution

It introduces a small energy regularity criterion for mean curvature flow applicable in arbitrary dimensions and codimensions, linking energy bounds to singularity prevention.

Findings

Small parabolic integral of |A|^2 prevents singularity formation.

The singular set of the flow has zero 2-dimensional Hausdorff measure.

Provides a criterion for regularity based on energy estimates.

Abstract

In this paper, we will derive a small energy regularity theorem for the mean curvature flow of arbitrary dimension and codimension. It says that if the parabolic integral of $|A|^2$ around a point in space-time is small, then the mean curvature flow cannot develop singularity at this point. As an application, we can prove that the 2-dimensional Hausdorff measure of the singular set of the mean curvature flow from a surface to a Riemannian manifold must be zero.