Open manifolds with asymptotically nonnegative Ricci curvature and large volume growth

TL;DR

This paper investigates the topology of open manifolds with asymptotically nonnegative Ricci curvature and large volume growth, establishing conditions under which they have finite topological types and extending results to manifolds with k-th asymptotically nonnegative Ricci curvature.

Contribution

It introduces new conditions linking curvature decay and volume growth to finite topological types, extending previous results to k-th asymptotically nonnegative Ricci curvature.

Findings

Manifolds with specified curvature decay and volume growth have finite topological type.

Extension of results to manifolds with k-th asymptotically nonnegative Ricci curvature.

Use of Abresch-Gromoll's excess function estimate in the generalization.

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 generize it to the manifolds with $k$-th asymptotically nonnegative Ricci curvature by using extensions of Abresch-Gromoll's excess function estimate.