A general regularity theory for weak mean curvature flow

TL;DR

This paper presents a new proof of partial regularity for weak mean curvature flow solutions, extending classical results to more general flows with background velocity fields using advanced analytical techniques.

Contribution

It introduces a novel proof method that generalizes Allard's regularity theorem to parabolic flows with background velocity fields in a sharp integrability class.

Findings

Proof extends to flows with background velocity fields

Achieves regularity up to C^{1,\varsigma}

Generalizes Allard's theorem to parabolic setting

Abstract

We give a new proof of Brakke's partial regularity theorem up to C^{1,\varsigma} for weak varifold solutions of mean curvature flow by utilizing parabolic monotonicity formula, parabolic Lipschitz approximation and blow-up technique. The new proof extends to a general flow whose velocity is the sum of the mean curvature and any given background flow field in a dimensionally sharp integrability class. It is a natural parabolic generalization of Allard's regularity theorem in the sense that the special time-independent case reduces to Allard's theorem.