In 1998 Smoczyk [Smo98] showed that, among others, the blowup limits at singularities are convex for the mean curvature flow starting from a closed star-shaped surface in $\mathbf{R}^3$. We prove in this paper that this is true for the mean curvature flow of star-shaped hypersurfaces in $\mathbf{R}^{n+1}$ in arbitrary dimension $n\geq 2$. In fact, this holds for a much more general class of initial hypersurfaces. In particular, this implies that the mean curvature flow of star-shaped hypersurfaces is generic in the sense of Colding-Minicozzi [CM12].

中文翻译：

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星形超曲面的平均曲率流

1998 年，Smoczyk [Smo98] 表明，对于从 $\mathbf{R}^3$ 中的闭合星形表面开始的平均曲率流，奇点处的爆破极限是凸的。我们在本文中证明，对于任意维度 $n\geq 2$ 中 $\mathbf{R}^{n+1}$ 中星形超曲面的平均曲率流是正确的。事实上，这适用于更一般的初始超曲面类。特别是，这意味着星形超曲面的平均曲率流在 Colding-Minicozzi [CM12] 的意义上是通用的。