On sufficient conditions to extend huber’s finite connectivity theorem to higher dimensions

Let M be a connected complete noncompact -dimensional Riemannian manifold with a base point p ∈ 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 ≥ 2. Note here that our radial curvatures can change signs wildly. We then show that lim_t→∞ vol B_t (p)/tⁿ 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.