Metrics of Negative Ricci Curvature

One of the most natural and important topics in Riemannian geometry is the relation between curvature and global structure of the underlying manifold. For instance, complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic. Furthermore, it seemed to be a natural principle that a (closed) manifold cannot carry two metrics of different signed curvatures, as it is a basic fact that this is true for sectional curvature. But it turned out to be wrong (much later and from a strongly analytic argument) for the scalar curvature S, since each manifold M', n > 3, admits a complete metric with S _-1 (cf. Aubin A and Bland, Kalka BIK). Hence the situation for Ricci curvature Ric, lying between sectional and scalar curvature, seemed to be quite delicate. Up to now, the most general results concerning Ric < 0 were proved by Gao, Yau GY and Brooks Br using Thurston's theory of hyperbolic threemanifolds, viz.: Each closed three-manifold admits a metric with Ric < 0. This is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities; Gao and Yau (resp. Brooks) smoothed these singularities to get a regular metric with Ric < 0. These methods extend to three-manifolds of finite type and certain hyperbolic orbifolds. In any case, the arguments rely on exploiting some extraordinary metric structures, whose existence is neither obvious nor conceptually related to the Ricci curvature problem. Indeed, the existence depends on the assumption that the manifold is three-dimensional and compact. Moreover this approach does not provide insight into the typical behaviour of metrics with Ric < 0 since one is led to very special metrics. In this article we approach negative Ricci curvature using a completely different and new concept (which will become even more significant in L2) as we deliberately produce Ric < 0. Actually we will prove the following results; in these notes Ric (g), resp. r (g), denotes the Ricci tensor, resp. curvature of a smooth metric g: