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

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 $n\ge 2$. 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 $M$.

8 pages, minor corrections