Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature (Q2804436)

The authors study minimal hypersurfaces in complete Riemannian manifolds that satisfy the following three conditions: (C1) non-negative Ricci curvature; (C2) Euclidean volume growth; (C3) quadratic decay of the curvature tensor.NEWLINENEWLINEThe main result is Theorem 5.5: Let \(N\) be an \((n+1)\)-dimensional complete Riemannian manifold satisfying conditions (C1), (C2) and (C3), and with non-radial Ricci curvature \(\inf _{\partial B_r}\text{Ric}(\xi^T,\xi^T)\geq k'r^{-2}\) almost everywhere for a constant \(k'\) and sufficiently large \(r>0\), where \(\xi^T\) stands for the part that is tangential to the geodesic sphere \(\partial B_r\) (at least away from the cut locus of the center), of a tangent vector \(\xi\) of \(N\) at the considered point. If \(k'>\frac{(n-2)^2}{4}\), then \(N\) admits no complete stable minimal hypersurfaces with at most Euclidean volume growth.NEWLINENEWLINEAnother works directly connected to this topic are [the first author, Chin. Ann. Math., Ser. B 32, No. 1, 27--44 (2011; Zbl 1211.53084)], [the second author and \textit{H. Karcher}, Manuscr. Math. 40, 27--77 (1982; Zbl 0502.53036)], and [the last author, Minimal submanifolds and related topics. River Edge, NJ: World Scientific (2003; Zbl 1055.53047)].