The construction of negatively Ricci curved manifolds

Abstract

The purpose of this paper is to construct complete Riemannian Manifolds with negative Ricci curvature metric. We shall mostly concentrate on threedimensional manifolds. In [BJ], Bourguignon raised the problem that the connected sum of two Riemannian Manifolds with negative Ricci curvature has a metric with negative Ricci curvature. Yau first observed that the connected sum of two hyperbolic manifolds has a negative Ricei curvature metric. In the first part of the paper, we shall give the affirmative answer to the above question. We actually prove something more. We shall prove that the "connected sum along circles" of two Riemannian Manifolds with negative Ricci curvature metric (Sect. 1) has a negative Ricci curvature metric. In the second part, we only consider three-dimensional manifolds. We use the above results and some topological constructions to obtain new manifolds with negative Ricci curvature metric from a given manifold with negative Ricci curvature metric. Give M a complete Riemannian three-dimensional manifold with negative Ricci curvature metric, then there are complete negative Ricci curvature Riemannian metric on M # S 2 x S I ; M ~ S 2 x S I # L ( p , q ) , and M ~ S 2 x S ~ 1; x S 1, where 2; is any oriented Riemann surface. For the motivation of this paper see [YS]. This paper is an extended version of the second part of the author's Ph.D. thesis at Stony Brook. I would like to thank my advisor Professor Blaine Lawson for suggesting this problem to me, and inspiring discussions and valuable advice.