A second derivative H\"{o}lder estimate for weak mean curvature flow

TL;DR

This paper proves that under certain conditions, weak mean curvature flow is smooth almost everywhere, and establishes C^{2,α} regularity when the velocity includes an α-Hölder continuous vector field.

Contribution

It provides a new regularity result for weak mean curvature flow, extending smoothness to almost all points under specific velocity conditions.

Findings

Brakke's mean curvature flow is smooth almost everywhere.

C^{2,α} regularity holds under weak velocity conditions with an α-Hölder continuous vector field.

The proof applies to flows with velocity equal to mean curvature plus a Hölder continuous vector field.

Abstract

We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given \alpha-H\"{o}lder continuous vector field, then we show C^{2,\alpha} regularity almost everywhere.