Minimizing currents in open manifolds and the n - 1 homology of nonnegatively Ricci curved manifolds

Abstract

We first of all obtain a condition under which a homology class of a complete noncompact Riemannian manifold can be represented by an area-minimizing integral current. We then use this result to study the codimension 1 integral homology of complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Specifically, we show the following two results. On one hand, if such an n dimensional manifold M has volume growth faster than 1, then the n - 1 homology group of M is trivial. If on the other hand the diameter growth of a similar M is bounded, then the n - 1 homology group is either trivial, Z , or Z 2 . We also offer other applications of the techniques developed for these theorems to both n - 1 homology groups and volume growth of minimal submanifolds in various complete noncompact Riemannian spaces.