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

A new version of Brakke's local regularity theorem