Vanishing of homology groups, Ricci estimate for submanifolds and applications

Abstract

In this paper we obtain an estimate for the Ricci curvature and a criterion for the vanishing of the homology groups of compact submanifolds of spheres and Euclidean spaces. This criterion depends on the results of Lawson and Simons [LS], Leung [Le2] and Xin [X] on stable currents. As consequences we obtain a topological version of theorems of Cheng and Nakagawa [CK], Alencar and do Carmo [AC], and Xu [Xu], on hypersurfaces in spheres with constant mean curvature.