On complete manifolds with nonnegative Ricci curvature

Abstract

Complete open Riemannian manifolds (Mn, g) with nonnegative sectional curvature are well understood. The basic results are Toponogov's Splitting Theorem and the Soul Theorem [CG1]. The Splitting Theorem has been extended to manifolds of nonnegative Ricci curvature [CG2]. On the other hand, the Soul Theorem does not extend even topologically, according to recent examples in [GM2]. A different method to construct manifolds which carry a metric with Ric > 0, but no metric with nonnegative sectional curvature, has been given by L. Berard Bergery [BB]. This leads to the question (cf. also [Y1]): Is there any finiteness result for complete Riemannian manifolds with Ric > 0 ? The answer is certainly affirmative in the low-dimensional special cases n = 2, where all notions of curvature coincide, and n = 3, where nonnegative Ricci curvature has been studied by means of stable minimal surfaces [MSY, SY]. On the other hand, J. P. Sha and D. G. Yang [ShY] have constructed complete manifolds with strictly positive Ricci curvature in higher dimensions. For example they can choose the underlying space to be R4 x S3 with infinitely many copies of S3 x CP 2 attached to it by surgery; cf. also [ShY 1]. It is therefore clear that any finiteness result for arbitrary dimensions requires additional assumptions. The purpose of this paper is to establish the following main result.