Frankel's property for free boundary minimal hypersurfaces in the Riemannian Schwarzschild manifolds

TL;DR

This paper investigates free boundary minimal hypersurfaces in Schwarzschild manifolds, proving intersection properties and examining effects of metric perturbations on scalar curvature.

Contribution

It establishes conditions for intersections of minimal hypersurfaces and analyzes the impact of metric perturbations on scalar curvature in Schwarzschild manifolds.

Findings

Free boundary minimal hypersurfaces must intersect certain hyperplanes under specific conditions.

The intersection occurs when the minimal hypersurface and hyperplane are closest in a bounded region.

Perturbations of the Schwarzschild metric can alter scalar curvature properties.

Abstract

We study the behavior of minimal hypersurfaces in the Schwarzschild $n$-manifolds that intersect the horizon orthogonally along the boundary. We show that a free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region. We also discuss when the Schwarzschild metric is perturbed in a way that its scalar curvature is no longer positive.