论文标题
关于$ s^k $的最小超曲面的交集
On the intersection of minimal hypersurfaces of $S^k$
论文作者
论文摘要
自从弗兰克尔（Frankel）的工作中，人们知道两个紧凑型沉浸式的最小超曲面在具有阳性RICCI曲率的歧管中必须具有相交点。在文献中可以找到这种结果的几种概括，例如在劳森，彼得森和威廉等人的作品中。在$ s^k $的最小曲面的特殊情况下，我们证明了弗兰克尔定理的更强版本。也就是说，我们表明，如果给出了两个紧凑的最小hypersurfaces $ m_1 $，$ m_2 $ $ s^k $和a点$ \ mathbf {p} \ in s^k $，则$ m_1 $ and $ m_2 $在$ \ mathbf {p} $方面具有半个峰值。作为该结果的必然，我们为一般尺寸案例提供了$ s^3 $的最小表面的ROS的两件式属性的替代证明。
It is known since the work of Frankel that two compactly immersed minimal hypersurfaces in a manifold with positive Ricci curvature must have an intersection point. Several generalizations of this result can be found in the literature, for example in the works of Lawson, Petersen and Wilhelm, among others. In the special case of minimal hypersurfaces of $S^k$, we prove a stronger version of Frankel's theorem. Namely, we show that if two compact minimal hypersurfaces $M_1$, $M_2$ of $S^k$ and a point $\mathbf{p}\in S^k$ are given, then $M_1$ and $M_2$ have an intersection point in the hemisphere with respect to $\mathbf{p}$. As a corollary of this result, we give an alternative proof to Ros' two-piece property of minimal surfaces of $S^3$, for the general dimension case.
