Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set.

*(English)*Zbl 0728.53026Let \((M^ n,o)\) be a Riemannian manifold with base point o. Two rays \(c_ 1\), \(c_ 2\) starting from o are called cofinal if for any \(r>0\) and \(t\geq r\), \(c_ 1(t)\) and \(c_ 2(t)\) are in the same component of M- B(o,r). An end is an equivalence class of cofinal rays. The author gives a universal bound for the number of ends of a Riemannian manifold with nonnegative Ricci curvature outside a geodesic ball B(o,a) and bounded from below on B(o,a). The key step of the proof is that for two different ends \([c_ 1]\), \([c_ 2]\) one has \(d(c_ 1(4a),c_ 2(4a))>2a\). This inequality combined with the Bishop-Gromov volume comparison theorem leads to the announced bound; however it is by no means sharp. The bound can be improved using a more general volume comparison theorem that the author only states in the present note.

Reviewer: L.Ornea (Bucureşti)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

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\textit{M. Cai}, Bull. Am. Math. Soc., New Ser. 24, No. 2, 371--377 (1991; Zbl 0728.53026)

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