On sufficient conditions to extend Huber's finite connectivity theorem to higher dimensions (Q2075406)

Let \(\left( M,g\right) \) be a complete connected noncompact \(n\)-dimensional Riemannian manifold, \(n\geq 2\), with a base point \(p\in M\). Let \(\left( M_{r},g_{r}\right) \) be a two-dimensional noncompact surface of revolution with a base point \(p^{\prime }\in M_{r}\). Then the Riemannian manifold \( \left( M,g\right) \) with the base point \(p\) is said to have its radial curvature bounded from below by that of the surface of revolution \(\left( M_{r},g_{r}\right) \) with the base point \(p^{\prime }\in M_{r}\) if, for every unit speed minimal geodesic \(\gamma :[0,a)\rightarrow M,\gamma \left( 0\right) =p\), the inequality \(K_{M}\left( \gamma ^{\prime }\left( t\right) ,v\right) \geq K_{r}\left( t\right) \) holds for every \(0\leq t<a\) and all \( v\in M_{\gamma \left( t\right) }\) which are orthogonal to \(\gamma ^{\prime }\left( t\right) \). Here, \(K_{M}\left( \gamma ^{\prime }\left( t\right) ,v\right) \) and \(K_{r}\left( t\right) \) are the sectional curvature and the radial curvature function of \(\left( M,g\right) \) and \(\left( M_{r},g_{r}\right) \), respectively. Let \(B_{t}\left( p\right) \) denote the open ball in \(\left( M,g\right) \) centered at \(p\) and of radius \(t>0\) and let \(\mathrm{Vol}\left( B_{t}\left( p\right) \right) \) denote its volume. The main goal of the paper is an extension of \textit{A. Huber}'s finite connectivity theorem [Comment. Math. Helv. 32, 13--72 (1957; Zbl 0080.15001)] to higher dimensions. The authors' main theorem (Theorem 1.2) states that if a complete connected noncompact \(n\)-dimensional Riemannian manifold \(\left( M,g\right) \), \(n\geq 2\), with a base point \(p\in M\) has its radial curvature at \(p\) bounded from below by that of a (two-dimensional) noncompact surface of revolution \(\left( M_{r},g_{r}\right) \) with a base point \(p^{\prime }\in M_{r}\) and such that \(\int_{M_{r}}K_{r-}dM_{r}>-\infty \), then \( \lim_{t\rightarrow \infty }\mathrm{Vol}\left( B_{t}\left( p\right) \right) /t\) exists; in addition, if this limit is not zero, then \(M\) has finite topological type and there is a finite upper bound on the number of ends of \( M\).