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


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DOI: https://doi.org/10.1090/S0002-9939-1995-1213856-8
Article copyright: © Copyright 1995 American Mathematical Society