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Boundaries of non-compact harmonic manifolds

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Abstract

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson–Sullivan measure and this measure coincides with the harmonic measure. As an application of these results we prove that the geodesic flow on a non-flat finite volume harmonic manifold without conjugate points is topologically transitive.

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Acknowledgments

I would like to thank Ralf Spatzier for many helpful conversations. I would also like to thank the referee for many helpful suggestions and corrections.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Andrew M. Zimmer

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  1. Andrew M. Zimmer
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Correspondence to Andrew M. Zimmer.

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The author thanks the National Science Foundation for support through the grant DMS-0602191.

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Zimmer, A.M. Boundaries of non-compact harmonic manifolds. Geom Dedicata 168, 339–357 (2014). https://doi.org/10.1007/s10711-013-9833-6

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