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, Leung, and Xin on stable currents. As consequences we obtain a topological version of theorems of Cheng and Nakagawa, Alencar and do Carmo, and Xu, on hypersurfaces in spheres with constant mean curvature.

In this paper, we shall prove the Nash inequality for a compact manifold with boundary with respect to weighted measure, using covering arguments of Jerison and Oden-Sung-Wang. We shall also state some results which are easily obtained from that inequality.

Using the idea of weighted sharing we prove a result on uniqueness of meromorphic functions sharing three values which improve some results of Ueda, Yi and Ye.