Rate of convergence of the mean curvature flow
Rate of convergence of the mean curvature flow
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平均曲率流的收敛率
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发表时间：
2005
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通讯作者：
N. Šešum
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作者：
N. Šešum
本文研究了光滑严格凸超曲面在n + 1中的平均曲率流Mt。表面保持光滑和凸，单调收缩，直到它在临界时间T和点x* 消失（这是由于Huisken）。这相当于说相应的重标度平均曲率流收敛到半径为Sn的球面Sn。本文主要研究重标流的指数收敛速度。我们将在这里提出一种方法，告诉我们指数衰减的速率至少是2/n。我们可以定义一个光滑的、严格凸的n维超曲面的“到达时间”u，当它以等于其平均曲率的法向速度运动时，通过u（x）= t，如果x ∈ Mt，其中x ∈ Int（M0）。Huisken证明了，对于n ≥ 2，u（x）在x* 附近是C2。Kohn和Serfaty [11]已经处理了n = 1的情况;他们证明了u的C3正则性。作为平均曲率流的收敛速度的结果，我们证明了当n ≥ 2时，u在x* 附近不一定是C3。我们还表明，所获得的收敛速度2/n，这是由线性化的平均曲率流，是最佳的，至少对n ≥ 2。© 2007 Wiley Periodicals，Inc.
We study the flow Mt of a smooth, strictly convex hypersurface by its mean curvature in ℝn + 1. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x* (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere Sn of radius √n. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us that the rate of the exponential decay is at least 2/n. We can define the “arrival time” u of a smooth, strictly convex, n‐dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u(x) = t if x ∈ Mt for x ∈ Int(M0). Huisken proved that, for n ≥ 2, u(x) is C2 near x*. The case n = 1 has been treated by Kohn and Serfaty [11]; they proved C3‐regularity of u. As a consequence of the obtained rate of convergence of the mean curvature flow, we prove that u is not necessarily C3 near x* for n ≥ 2. We also show that the obtained rate of convergence 2/n, which arises from linearizing a mean curvature flow, is the optimal one, at least for n ≥ 2. © 2007 Wiley Periodicals, Inc.