On minimal hypersurfaces of nonnegatively Ricci curved manifolds
Let \(M\) be a complete, connected noncompact \(d\)-dimensional Riemannian manifold having everywhere nonnegative Ricci curvature. In this paper a compact hypersurface \(\Sigma\) of Hausdorff dimension \(d- 1\), is an integral current. Then the support of \(\Sigma\), \(\text{Spt }\Sigma\), is (almost everywhere) the union of a finite number of Lipschitz images in \(M\) of compact \((d- 1)\)-dimensional orientable Riemannian manifolds. Assume \(\Sigma\) without boundary and locally minimizing (in a precise sense described in the paper). The main result of the paper can be described as follows: for some covering space \(p: \widetilde M:\to M\), there exists a smooth compact orientable \((d- 1)\)-dimensional manifold \(T\) and a (possible empty) compact region \(Q\subseteq \widetilde M\) such that \(\widetilde M- Q\) splits isometrically as either the Riemannian product \(T\times R\) or \(T\times [0,\infty)\) and \(\text{Spt }p^*\Sigma\) consists of disjoint smoothly embedded copies of \(T\). The author mentions that this generalizes a previously known result, where \(\Sigma\) is a compact smooth \((d- 1)\)-dimensional minimal manifold.
scientific article; zbMATH DE number 436740