Partial Regularity of Mean-Convex Hypersurfaces Flowing by Mean Curvature
نویسنده
چکیده
In this paper we announce various new results about singularities in the mean curvature flow. Some results apply to any weak solution (i.e., any Brakke flow of integral varifolds.) Our strongest results, however, are for initially regular mean-convex hypersurfaces. (We say a hypersurface is mean-convex if it bounds a region such that the mean curvature with respect to the inward unit normal is positive at each point of the surface.) In this case we can prove the optimal result about the Hausdorff dimension of the singular set in spacetime. The results hold in fairly general ambient manifolds (all compact riemannian manifolds, for example), but in this note we describe only surfaces in euclidean space. The full results use the deep Brakke regularity theorems [B]. However, if one only cares about the flow up to and including the first time at which singularities appear, then Brakke’s regularity theorems are not needed. Our methods also yield elementary new proofs of the theorems of Huisken [H1], Gage-Hamilton [GH], and Grayson [G1], [G2] about convex hypersurfaces, convex curves, and embedded curves. In this paper, X, Y, . . . denote points in spacetime R × Rn+1, and x, y, . . . denote points in space (Rn+1). We let T (X) and S(X) be the time and space components, respectively, ofX, so thatX = (T (X), S(X)). We make spacetime into a metric space by letting the parabolic distance between X and Y be ‖X − Y‖ = max{|T (X) − T (Y)|1/2, |S(X) − S(Y)|}. The parabolic Hausdorff dimension of a set in spacetime means Hausdorff dimension with respect to this metric. Thus the parabolic Hausdorff dimension of any subset of a given time slice T−1(t) is the same as its ordinary euclidean Hausdorff dimension, whereas the time axis (0) × Rn+1 has parabolic Hausdorff dimension 2.
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