Open manifolds with asymptotically nonnegative Ricci curvature and large volume growth

Abstract

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume growth conditions. We also generalize it to the manifolds with kth asymptotically nonnegative Ricci curvature by using extensions of Abresch-Gromoll's excess function estimate.

Sobolev Inequalities In Manifolds With Asymptotically Nonnegative Curvature

“…Some important geometric, topological and analysis problems have been investigated for this kind of manifolds (cf. [2], [3], [21,22], [27,26], [37], [18], [6], [36], etc). Now we recall its definition as follows.…”