A new version of Brakke's local regularity theorem

Abstract

Consider an integral Brakke flow $(\mu_t)$, $t\in [0,T]$, inside some ball in Euclidean space. If $\mu_{0}$ has small height, its measure does not deviate too much from that of a plane and if $\mu_{T}$ is non-empty, then Brakke's local regularity theorem yields that $(\mu_t)$ is actually smooth and graphical inside a smaller ball for times $t\in (C,T-C)$ for some constant $C$. Here we extend this result to times $t\in (C,T)$. The main idea is to prove that a Brakke flow that is initially locally graphical with small gradient will remain graphical for some time. Moreover we use the new local regularity theorem to generalise White's regularity theorem to Brakke flows.