On the intersection of minimal hypersurfaces of $S^k$

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.

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