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Article Dans Une Revue Mathematische Zeitschrift Année : 2009

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

Résumé

The Liouville property of a complete Riemannian manifold (i.e., the question whether there exist non-trivial bounded harmonic functions) 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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Dates et versions

hal-00245623 , version 1 (07-02-2008)

Identifiants

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Marc Arnaudon, Anton Thalmaier, Stefanie Ulsamer. Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature. Mathematische Zeitschrift, 2009, 263, pp.369-409. ⟨10.1007/s00209-008-0422-6⟩. ⟨hal-00245623⟩

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