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

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Abstract: Let be a pointed open complete manifold with Ricci curvature bounded from below by (for ) and nonnegative outside the ball . It has recently been shown that there is an upper bound for the number of ends of such a manifold which depends only on and the dimension *n* of the manifold . We will give a gap theorem in this paper which shows that there exists an such that has at most two ends if . We also give examples to show that, in dimension , such manifolds in general do not carry any complete metric with nonnegative Ricci Curvature for any .

**[A]**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****[AG]**Uwe Abresch and Detlef Gromoll,*On complete manifolds with nonnegative Ricci curvature*, J. Amer. Math. Soc.**3**(1990), no. 2, 355–374. MR**1030656**, 10.1090/S0894-0347-1990-1030656-6**[C]**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**, 10.1090/S0273-0979-1991-16038-6**[CG]**Jeff Cheeger and Detlef Gromoll,*The splitting theorem for manifolds of nonnegative Ricci curvature*, J. Differential Geometry**6**(1971/72), 119–128. MR**0303460****[EH]**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**, 10.1007/BF01876506**[L]**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**, 10.1090/S0002-9939-1992-1068127-7**[LT]**P. Li and F. Tam,*Harmonic functions and the structure of complete manifolds*, preprint, 1990.**[SY]**Ji-Ping Sha and DaGang Yang,*Positive Ricci curvature on the connected sums of 𝑆ⁿ×𝑆^{𝑚}*, J. Differential Geom.**33**(1991), no. 1, 127–137. MR**1085137**

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1213856-8

Article copyright:
© Copyright 1995
American Mathematical Society