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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, than 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.