End-time regularity theorem for Brakke flows

Abstract

For a general $k$-dimensional Brakke flow in ${\mathbb{R}}^n$ locally close to a $k$-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the end-time. Moreover, at any point in space-time where the Gaussian density is close to $1$, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White's local regularity theorem to general Brakke flows. The regularity result is in fact obtained for more general Brakke-like flows, driven by the mean curvature plus an additional forcing term in a dimensionally sharp integrability class or in a Hölder class.

29 pages, 1 figure. In v2 we have added some further explanations of a few technical details. This is the final version, to appear on Mathematische Annalen

References in corpus (2)

• On the existence of canonical multi-phase Brakke flows
• A short proof of Allard's and Brakke's regularity theorems

Cited by in corpus (1)

• Epsilon-regularity for the Brakke flow with boundary