Let M be a connected complete noncompact n-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 limt→∞ vol Bt(p)/tn exists where vol Bt(p) denotes the volume of the open metric ball Bt(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.
MSC Codes Primary 53C20, 53C21, Secondary 53C22, 53C23
|Publication status||Published - Dec 24 2019|
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