TAILIEUCHUNG - Đề tài "A local regularity theorem for mean curvature flow "

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 | Annals of Mathematics A local regularity theorem for mean curvature flow By Brian White 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 embedded 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 Theorem. There are numbers e e N 0 and C C N TO with the following property. If M is a smooth proper mean curvature flow in an open subset U of the spacetime RN X R and if the Gaussian density ratios of M are bounded above by 1 e then at each spacetime point X x t of M the norm of the second fundamental form of M at X is bounded by C Ỗ X U where Ỗ X U is the infimum of X YII among all points Y y s G Uc 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 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. 1488 BRIAN WHITE 1 The theory developed here applies up to and including the time at which singularities first occur in any classical mean curvature flow. See Theorem . 2 The bounds carry over easily to any varifold flow that is a weak limit of smooth mean curvature flows. See 7. In particular any smooth compact embedded hypersurface of RN is the initial surface of such a flow for 0 t TO . 3 The bounds also extend .

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