Partial regularity of mean-convex hypersurfaces flowing by mean curvature
Partial regularity of mean-convex hypersurfaces flowing by mean curvature
复制标题
平均曲率流动的平均凸超曲面的部分正则性
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发表时间：
1994
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通讯作者：
B. White
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文献类型：
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作者：
B. White
在本文中，我们公布了关于平均曲率流中奇点的各种新结果。一些结果适用于任何弱解(即，任何积分变差的Brakke流)。然而，我们最强的结果是关于初始正则的平均凸超曲面。(我们说一个超曲面是平均凸的，如果它限定了一个区域，使得相对于向内单位法线的平均曲率在曲面的每个点处都是正的。)在这种情况下，我们可以证明时空中奇异集的Hausdorff维数的最优结果。这些结果适用于相当一般的环境流形(例如，所有紧致黎曼流形)，但在本文中，我们仅描述欧氏空间中的曲面。全部结果使用了深入的Brakke正则性定理[B]。然而，如果一个人只关心直到奇点出现的第一次的流动，那么就不需要布拉克的正则性定理。我们的方法也给出了Huisken[H1]、Gage-Hamilton[GH]和Grayson[G1]、[G2]关于凸超曲面、凸曲线和嵌入曲线的定理的新的初等证明。在本文中，X，Y，.。。表示时空R×Rn+1中的点和x，y，.。。表示空间(Rn+1)中的点。设T(X)和S(X)分别是X的时间分量和空间分量，使得X=(T(X)，S(X))。设X和Y之间的抛物线距离为‖X−Y‖=max{|T(X)−T(Y)|1/2，|S(X)−S(Y)|}，使时空进入度规空间。时空中集合的抛物线Hausdorff维是指关于该度规的Hausdorff维。因此，给定时间片T−1(T)的任何子集的抛物线Hausdorff维与其普通欧几里德Hausdorff维相同，而时间轴(0)×Rn+1具有抛物线Hausdorff维2。
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, of X, so that X = (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.