极小超曲面的半空间相交性质
Half-space intersection properties for minimal hypersurfaces
AI总结：
本文在半球、空间形式半测地球和高斯空间子集三种情形下证明极小超曲面的半空间相交性质，并给出收缩子相交半径上界$2\sqrt{n}$。
摘要：
我们在三种情形下证明了“半空间”相交性质：半球、空间形式中的半测地球以及高斯空间的某些子集。例如，球面中任意两个嵌入的极小超曲面必须在每个闭半球中相交。我们发展了两种方法：一种利用稳定极小超曲面的分类，另一种利用共形变换和关于$\alpha$-Bakry-Émery-Ricci曲率的比较几何。我们的方法得到了空间形式球中自由边界极小超曲面的类似相交性质，即使内部或边界曲率可能为负。最后，Colding和Minicozzi最近证明了任意两个维数为$n$的嵌入收缩子必须在半径足够大的欧几里得球$R(n)$中相交。我们证明$R(n) \leq 2 \sqrt{n}$。
英文摘要：
We prove ``half-space" intersection properties in three settings: the hemisphere, half-geodesic balls in space forms, and certain subsets of Gaussian space. For instance, any two embedded minimal hypersurfaces in the sphere must intersect in every closed hemisphere. Two approaches are developed: one using classifications of stable minimal hypersurfaces, and the second using conformal change and comparison geometry for $α$-Bakry-Émery-Ricci curvature. Our methods yield the analogous intersection properties for free boundary minimal hypersurfaces in space form balls, even when the interior or boundary curvature may be negative. Finally, Colding and Minicozzi recently showed that any two embedded shrinkers of dimension $n$ must intersect in a large enough Euclidean ball of radius $R(n)$. We show that $R(n) \leq 2 \sqrt{n}$.
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