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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Martin points on open manifolds of non-positive curvature


Authors: Jianguo Cao, Huijun Fan and François Ledrappier
Journal: Trans. Amer. Math. Soc. 359 (2007), 5697-5723
MSC (2000): Primary 58J32; Secondary 60J65
Published electronically: July 3, 2007
MathSciNet review: 2336303
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Abstract: The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact manifold with Ballmann rank one, we show that Martin points are generic and of full harmonic measure. The result of this paper provides a partial answer to an open problem of S. T. Yau.


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Additional Information

Jianguo Cao
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: jcao@nd.edu

Huijun Fan
Affiliation: School of Mathematical Sciences, Peking University, Beijing 100875, People’s Republic of China
Email: fanhj@math.pku.edu.cn

François Ledrappier
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: fledrapp@nd.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04102-5
PII: S 0002-9947(07)04102-5
Received by editor(s): May 27, 2005
Published electronically: July 3, 2007
Additional Notes: The first author was supported in part by an NSF grant
The second author was partially supported by the Research Fund for returned overseas Chinese Scholars 20010107 and by the NSFC for Young Scholars 10401001
Article copyright: © Copyright 2007 American Mathematical Society