A Volume Comparison Theorem for Manifolds with Asymptotically Nonnegative Curvature and its Applications

Abstract

where dpo is the distance to Po. This class was first studied by U. Abresch ([AbI], [Ab2]). Because of the decay condition on the curvature (faster than quadratic), we expect the class of manifolds of asymptotically nonnegative curvature behaves similarly at infinity to manifolds of nonnegative curvature. For the case of sectional curvature, in [Abl] and [Ab2], Abresch showed that the complexity of topology of such manifolds is similar to that of nonnegative curvature, namely, they are all of finite topological type with uniform bound on the Betti numbers. In [Ka], A. Kasue investigated the theory of harmonic functions for such manifolds. Again, he demonstrated the expected similarity. For Ricci curvature, the examples of J.-P. Sha and D.-G. Yang showed that we can not hope to control the topology of such manifolds as we do for sectional curvature. Very little is known for the case of asymptotically nonnegative Ricci curvature except the work of Abresch and Gromoll ([AB]) which studied such manifolds under further conditions on the sectional curvature and diameter growth. A question of recent interest, stimulated by the works of P. Li and L.-F. Tam ([LT1], [LT2]), is whether open manifolds of asymptotically nonnegative Ricci curvature has finitely many ends. For the special case when A has compact support, M.-L. Cai ([Cai]) and Z.-D. Liu ([Liu]) recently gave a bound on the number of ends. In [LT2], under the stronger condition of f0o tn' A(t) dt < o0, Li and Tam also gave such a bound.