Counting ends on complete smooth metric measure spaces (Q2792146)

Let \((M^n,g,e^{-f}dv)\) be an \(n\)-dimensional complete smooth Riemannian manifold with a measure.NEWLINENEWLINEApplying the \(f\)-volume comparison theorem of \textit{G. Wei} and \textit{W. Wylie} [J. Differ. Geom. 83, No. 2, 377--405 (2009; Zbl 1189.53036)] and using the geometrical properties of geodesics, the author estimates the upper bound of the number of ends of the manifold \(M^n\) in the case where the (\(\infty\)-)Bakry-Émery Ricci curvature is nonnegative outside a compact set. If instead of the \(\infty\)-Bakry-Émery Ricci curvature we consider the \(m\)-one, this estimation is independent of the values of \(f\). The proof of the main result follows patterns similar to that in [\textit{M. Cai}, Bull. Am. Math. Soc., New Ser. 24, No. 2, 371--377 (1991; Zbl 0728.53026)].