Compactness criteria for Riemannian manifolds with compact unstable minimal hypersurfaces
The author proves an Ambrose-type theorem as follows: Let N be a complete Riemannian manifold with a compact embedded unstable minimal hypersurface M. Suppose that there exists a positive constants \(s_ 0\) such that along each unit speed geodesic \(\gamma\) : \([0,\infty)\to N\) emanating from each point in the tubular neighborhood \(U_{s_ 0}(M):=\{g\in N;\quad dist_ N(g,M)<s_ 0\}\) the Ricci curvature satisfies \(\liminf_{r\to \infty}\int^{r}_{0}Ric_ N(d\gamma /dt,d\gamma /dt)dt\geq 0.\) Then N is compact. For other Ambrose-type theorems see \textit{G. J. Galloway} [Proc. Am. Math. Soc. 84, 106-110 (1982; Zbl 0484.53032)].
scientific article; zbMATH DE number 4149105