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

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

2008-02-07

arxiv.org/abs/0802.0966v1

Mathematics

Probability

Scientific paper

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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