A BRAY-BRENDLE-NEVES TYPE INEQUALITY FOR A RIEMANNIAN MANIFOLD

In this paper, for any local area-minimizing closed hypersurface Σ with ■, immersed in a(n + 1)-dimension Riemannian manifold M which has positive scalar curvature and nonnegative Ricci curvature, we obtain an upper bound for the area of Σ. In particular, when Σ saturates the corresponding upper bound, Σ is isometric to Sⁿ and M splits in a neighborhood of Σ. At the end of the paper, we also give the global version of this result.     

A NOTE ON COMPLETE MANIFOLDS WITH FINITE VOLUME

In this article, we concern on complete manifolds with finite volume. We prove that under some assumptions about scalar curvature and the Yamabe constant, the manifolds must be compact, and we also give the diameter estimates in terms of the scalar curvature and the Yamabe constant.