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 $M$.
Dedication
Dedicated Professor K. Shiohama on his eightieth birthday
Citation
Kei Kondo. Yusuke Shinoda. "On sufficient conditions to extend Huber's finite connectivity theorem to higher dimensions." Tohoku Math. J. (2) 73 (3) 463 - 470, 2021. https://doi.org/10.2748/tmj.20200701
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