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Mathematics, Analysis of PDEs, math.AP,Mathematics, Differential Geometry, math.DG,
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.
