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Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature

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Abstract

The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Applications (UMR6086), Université de Poitiers, Téléport 2, BP 30179, 86962, Futuroscope Chasseneuil Cedex, France

    Marc Arnaudon

  2. Unité de Recherche en Mathématiques, Université du Luxembourg, 162A, avenue de la Faïencerie, 1511, Luxembourg, Grand-Duchy of Luxembourg

    Anton Thalmaier

  3. NWF I, Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Stefanie Ulsamer

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  1. Marc Arnaudon
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  2. Anton Thalmaier
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  3. Stefanie Ulsamer
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Correspondence to Anton Thalmaier.

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Arnaudon, M., Thalmaier, A. & Ulsamer, S. Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature. Math. Z. 263, 369–409 (2009). https://doi.org/10.1007/s00209-008-0422-6

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  • DOI: https://doi.org/10.1007/s00209-008-0422-6

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