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

by Marc Arnaudon, Anton Thalmaier and Stefanie Ulsamer

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.
*Math. Zeitschrift* **263** (2009) 369-409

The paper is available here:

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