Abstract:

We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^\{n-1\}$-estimates for the regularity scale of the level set flow with two-convex initial data. The results, which seem new even in the most classical case of mean convex surfaces evolving by mean curvature flow in $\\Bbb\{R\}^3$, are ultimately a consequence of the Lojasiewicz inequality from Colding-Minicozzi.

中文翻译：

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平均曲率流量下的直径和曲率控制

摘要：

我们证明，对于两个凸超曲面的平均曲率流，当一个接近第一个奇异时间时，内在直径保持一致。我们还用两个凸的初始数据得出了水平集流的规则性尺度的尖锐$ L ^ \ {n-1 \} $-估计。即使在最经典的情况下，平均凸曲面通过平均曲率流以\\\ Bbb \ {R \} ^ 3 $演化，结果似乎也是新的，这最终是Colding-Minicozzi的Lojasiewicz不等式的结果。