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Proceedings of the American Mathematical Society

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



Ball covering on manifolds with nonnegative Ricci curvature near infinity

Author: Zhong-dong Liu
Journal: Proc. Amer. Math. Soc. 115 (1992), 211-219
MSC: Primary 53C20; Secondary 53C21
MathSciNet review: 1068127
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Abstract: Let $ M$ be a complete open Riemannian manifold with nonnegative Ricci curvature outside a compact set $ B$. We show that the following ball covering property (see [LT]) is true provided that the sectional curvature has a lower bound:

For a fixed $ {p_0} \in M$, there exist $ N > 0$ and $ {r_0} > 0$ such that for $ r \geq {r_0}$, there exist $ {p_1},{\kern 1pt} \ldots ,{p_k} \in \partial {B_{p0}}(2r),k \leq N$ , with

$\displaystyle \bigcup\limits_{j = 1}^k {{B_{{p_j}}}(r) \supset \partial {B_{{p_0}}}(2r).} $

Furthermore $ N$ and $ {r_0}$ depend only on the dimension, the lower bound on the sectional curvature, and the radius of the ball at $ {p_0}$ that contains $ B$.

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

  • [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)
  • [BC] R. Bishop and R. Crittenden, Geometry of manifolds, Academic Press, New York, 1964. MR 0169148 (29:6401)
  • [CGT] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), 15-53. MR 658471 (84b:58109)
  • [GLP] M. Gromov, J. Lafontaine, and P. Pansu, Structures métriques pour les variétés Riemanniennes, Nathan, Paris, 1982.
  • [LT] P. Li and L. Tam, Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set, Ann. of Math. 125 (1987), 171-207. MR 873381 (88m:58039)
  • [LY] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201. MR 834612 (87f:58156)
  • [S] Z. Shen, Finiteness and vanishing theorems for complete open Riemannian manifolds, Bull. Amer. Math. Soc. 21 (1989), 241-244. MR 998628 (90c:53111)
  • [Y] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. MR 0431040 (55:4042)

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Keywords: Ricci curvature, end, volume comparison, geodesic ball
Article copyright: © Copyright 1992 American Mathematical Society

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