Rigidity of stable minimal hypersurfaces
In this interesting paper, the authors -- motivated by \textit{D. Fisher-Colbrie} and \textit{R. Schoen} [Common. Pure Appl. Math. 33, 199-211 (1980; Zbl 0439.53060)] and \textit{R. Schoen} and \textit{S.-T. Yau} [Ann. Math. Stud. 102, 209-228 (1982; Zbl 0481.53036)] -- study stability of minimal hypersurfaces assuming, on the ambient manifold, that its dimension is at least 4 and a natural condition on its sectional curvature, and the completeness of the hypersurface. Concretely, they prove that if \(\Sigma^n=N\times {\mathbb R}\) is a complete stable minimal hypersurface in \(M^{n+1}\), where \(N\) is a compact manifold admitting a metric of nonpositive sectional curvature, \(n=3\) or \(n=4\) and the sectional curvature of \(M\) is nonnegative, then \(\Sigma ^n\) is flat and totally geodesic. When \(n=2\) this result was proved in the paper cited above, under the nonnegative scalar curvature condition. The authors conjecture that this result can be true for all dimensions. Under similar conditions, they also prove a global splitting theorem for this case.
scientific article; zbMATH DE number 1058821