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Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

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Abstract

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature \(h\). In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds \(X\) with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon–Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification \(\overline{X}\).

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Acknowledgments

The authors would like to thank Evangelia Samiou for bringing the references [10, 11] to their attention.

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Authors and Affiliations

  1. Faculty of Mathematics, Ruhr University Bochum, 44780, Bochum, Germany

    Gerhard Knieper

  2. Department of Mathematical Sciences, Durham University, Durham, DH13LE, UK

    Norbert Peyerimhoff

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  1. Gerhard Knieper
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  2. Norbert Peyerimhoff
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Correspondence to Gerhard Knieper.

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Knieper, G., Peyerimhoff, N. Harmonic Functions on Rank One Asymptotically Harmonic Manifolds. J Geom Anal 26, 750–781 (2016). https://doi.org/10.1007/s12220-015-9570-1

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  • DOI: https://doi.org/10.1007/s12220-015-9570-1

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