Tohoku Mathematical Journal
2021

September
SECOND SERIES VOL. 73, NO. 3

Tohoku Math. J.
73 (2021), 463-470

Title ON SUFFICIENT CONDITIONS TO EXTEND HUBER'S FINITE CONNECTIVITY THEOREM TO HIGHER DIMENSIONS

Dedicated Professor K. Shiohama on his eightieth birthday

Author Kei Kondo and Yusuke Shinoda

(Received December 26, 2019, revised June 22, 2020)
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 $\lim_{t\to\infty}\vol B_t(p) / t^n$ exists where $\vol B_t(p)$ 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 $M$.

Mathematics Subject Classification. Primary 53C20; Secondary 53C21, 53C22, 53C23.

Key words and phrases. End, finite topological type, radial curvature, total curvature.

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