On sufficient conditions to extend Huber's finite connectivity theorem to higher dimensions

TL;DR

This paper extends Huber's finite connectivity theorem to higher-dimensional noncompact Riemannian manifolds with certain curvature bounds, establishing volume growth limits and topological finiteness under specified conditions.

Contribution

It provides sufficient conditions involving radial curvature bounds for higher-dimensional manifolds to have finite topological type and controlled volume growth.

Findings

The volume growth ratio limit exists for the manifold.

If the volume growth limit is positive, the manifold has finite topological type.

The number of ends of the manifold is finitely bounded.

Abstract

Let $M$ be a connected complete noncompact $n$-dimensional Riemannian manifold with a base point $p \in M$ whose radial sectional curvature at $p$ is bounded from below by that of a noncompact surface of revolution which admits a finite total curvature where $n \geq 2$. Note here that our radial curvatures can change signs wildly. We then show that $\lim_{t\to\infty} \mathrm{vol} B_t(p) / t^n$ exists where $\mathrm{vol} B_t(p)$ denotes the volume of the open metric ball $B_t(p)$ with center $p$ and radius $t$. Moreover we show that in addition if the limit above is positive, then $M$ has finite topological type and there is therefore a finitely upper bound on the number of ends of $M$.