Frankel's property for free boundary minimal hypersurfaces in the Riemannian Schwarzschild manifolds (Q6644839)

In the theory of general relativity, the Schwarzschild solution to an Einstein field equation is an important model in which one can describe the local geometry of spacetime in the solar system. It represents an empty spacetime outside the spherical mass, which also has spherical symmetry and asymptotically flat behavior. The Schwarzschild solution can be described as follows:\N\[\NM^{n} = \left\{x \in \mathbb{R}^{n} \mid \|x\| \geq \left(\tfrac{m}{2}\right)^{\frac{1}{n-2}}\right\}, \quad g_{Sch}^{n}=\left(1+\tfrac{m}{2\|x\|^{n-2}}\right)^{\frac{4}{n-2}}g_{\mathbb{R}^{n}}.\N\]\NHere, \(m\) is defined as the ADM mass of the manifold. The spherical mass body is called a horizon and it is a totally geodesic hypersurface in \(M^{n}\).\N\NOne of the most important aspects of the Riemannian Schwarzschild manifold is to study its minimal submanifolds. Indeed, the question was raised whether the horizon and totally geodesic planes are the only embedded minimal surfaces in \(M^{n}\).\N\NIn this paper, the authors prove that a free boundary minimal hypersurface and a totally geodesic hyperplane must intersect when the distance between them is achieved in a bounded region. As a consequence, if a smooth connected two-sided properly immersed minimal hypersurface \(\Sigma\) is on one side of a totally geodesic hyperplane \(P\) and the distance between them is achieved in a bounded region then \(\Sigma=P\).