A gap theorem for ends of complete manifolds
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- by Mingliang Cai, Tobias Holck Colding and DaGang Yang PDF
- Proc. Amer. Math. Soc. 123 (1995), 247-250 Request permission
Abstract:
Let $({M^n},o)$ be a pointed open complete manifold with Ricci curvature bounded from below by $- (n - 1){\Lambda ^2}$ (for $\Lambda \geq 0$) and nonnegative outside the ball $B(o,a)$. It has recently been shown that there is an upper bound for the number of ends of such a manifold which depends only on $\Lambda a$ and the dimension n of the manifold ${M^n}$. We will give a gap theorem in this paper which shows that there exists an $\varepsilon = \varepsilon (n) > 0$ such that ${M^n}$ has at most two ends if $\Lambda a \leq \varepsilon (n)$. We also give examples to show that, in dimension $n \geq 4$, such manifolds in general do not carry any complete metric with nonnegative Ricci Curvature for any $\Lambda a > 0$.References
- Uwe Abresch, Lower curvature bounds, Toponogov’s theorem, and bounded topology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 4, 651–670. MR 839689, DOI 10.24033/asens.1499
- Uwe Abresch and Detlef Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), no. 2, 355–374. MR 1030656, DOI 10.1090/S0894-0347-1990-1030656-6
- Mingliang Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 2, 371–377. MR 1071028, DOI 10.1090/S0273-0979-1991-16038-6
- Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119–128. MR 303460
- Jost Eschenburg and Ernst Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Global Anal. Geom. 2 (1984), no. 2, 141–151. MR 777905, DOI 10.1007/BF01876506
- Zhong-dong Liu, Ball covering on manifolds with nonnegative Ricci curvature near infinity, Proc. Amer. Math. Soc. 115 (1992), no. 1, 211–219. MR 1068127, DOI 10.1090/S0002-9939-1992-1068127-7 P. Li and F. Tam, Harmonic functions and the structure of complete manifolds, preprint, 1990.
- Ji-Ping Sha and DaGang Yang, Positive Ricci curvature on the connected sums of $S^n\times S^m$, J. Differential Geom. 33 (1991), no. 1, 127–137. MR 1085137
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 247-250
- MSC: Primary 53C20; Secondary 53C21
- DOI: https://doi.org/10.1090/S0002-9939-1995-1213856-8
- MathSciNet review: 1213856