Abstract
For a manifold with nonpositive curvature, the Martin boundary is described by the behavior of normalized Green’s functions at infinity. A classical result by Anderson and Schoen states that if the manifold has pinched negative curvature, the geometric boundary is the same as the Martin boundary. In this paper, we study the Martin boundary of rank 1 manifolds admitting compact quotients. It is proved that a residual set in the geometric boundary can be identified naturally with a subset of the Martin boundary. This gives a partial answer to one of the open problems in geometry collected by Yau.
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Acknowledgements
The author would like to thank Professor Józef Dodziuk for the invaluable suggestions and constant encouragement. The author is grateful to Professor Werner Ballmann for helpful discussions. The author is also grateful to the referees for their careful reading and important comments that significantly improved the exposition.
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Ji, R. On the Martin Boundary of Rank 1 Manifolds with Nonpositive Curvature. J Geom Anal 29, 2805–2822 (2019). https://doi.org/10.1007/s12220-018-0095-2
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DOI: https://doi.org/10.1007/s12220-018-0095-2