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Summary: hal-00245623,version1-7Feb2008
EXISTENCE OF NON-TRIVIAL HARMONIC FUNCTIONS
ON CARTAN-HADAMARD MANIFOLDS
OF UNBOUNDED CURVATURE
MARC ARNAUDON, ANTON THALMAIER, AND STEFANIE ULSAMER
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.
Contents
1. Introduction 1
2. Construction of a CH manifold with a sink of curvature at infinity 8
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