A Local Regularity Theorem for Mean Curvature Flow
Annals of Mathematics, 161 (2005), 1487–1519 A local regularity theorem for mean curvature flow By Brian White* Abstract This paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the Gaussian densities are close to 1. Introduction Let Mt with 0 < t < T be a smooth one-parameter family of embed- ded manifolds, not necessarily compact, moving by mean curvature in RN . This paper proves uniform curvature bounds in regions of spacetime where the Gaussian density ratios are close to 1. For instance (see 3.4): § Theorem. There are numbers ε = ε(N) > 0 and C = C(N) < with ∞ the following property. If is a smooth, proper mean curvature flow in an M open subset U of the spacetime RN R and if the Gaussian density ratios of × are bounded above by 1 + ε, then at each spacetime point X = (x, t) of , M M the norm of the second fundamental form of at X is bounded by M C δ(X, U) where δ(X, U) is the infimum of X Y among all points Y = (y, s) U c # − # ∈ with s t. ≤ (The terminology will be explained in 2.) § Another paper [W5] extends the bounds to arbitrary mean curvature flows of integral varifolds. However, that extension seems to require Brakke’s Local Regularity Theorem [B, 6.11], the proof of which is very difficult. The results of this paper are much easier to prove, but nevertheless suffice in many interesting situations. In particular: *The research presented here was partially funded by NSF grants DMS-9803403, DMS- 0104049, DMS-0406209 and by a Guggenheim Foundation Fellowship.
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