Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A gap theorem for ends of complete manifolds

Authors: Mingliang Cai, Tobias Holck Colding and DaGang Yang
Journal: Proc. Amer. Math. Soc. 123 (1995), 247-250
MSC: Primary 53C20; Secondary 53C21
MathSciNet review: 1213856
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A] U. Abresch, Lower curvature bounds, Toponogov's theorem and bounded topology, Ann. Sci. École Norm. Sup. (4) 18 (1985), 651-670. MR 839689 (87j:53058)
  • [AG] U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), 355-374. MR 1030656 (91a:53071)
  • [C] M. Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. Amer. Math. Soc. (N.S.) 24 (1991), 371-377. MR 1071028 (92f:53045)
  • [CG] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geom. 6 (1971), 119-128. MR 0303460 (46:2597)
  • [EH] J.-H. Eschenburg and E. Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Global Anal. Geom. 2 (1984), 249-260. MR 777905 (86h:53042)
  • [L] Z. Liu, Ball covering on manifolds with nonnegative Ricci curvature near infinity, Proc. Amer. Math. Soc. 115 (1992), 211-219. MR 1068127 (92h:53046)
  • [LT] P. Li and F. Tam, Harmonic functions and the structure of complete manifolds, preprint, 1990.
  • [SY] J. P. Sha and D. G. Yang, Positive Ricci curvature on the connected sums of $ {S^n} \times {S^m}$, J. Differential Geom. 33 (1991), 127-137. MR 1085137 (92f:53048)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C20, 53C21

Retrieve articles in all journals with MSC: 53C20, 53C21

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society