 
Summary: hal00245623,version17Feb2008
EXISTENCE OF NONTRIVIAL HARMONIC FUNCTIONS
ON CARTANHADAMARD 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 nontrivial bounded harmonic functions
on M) attracted a lot of attention. For CartanHadamard manifolds the role
of lower curvature bounds is still an open problem. We discuss examples of
CartanHadamard 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 nondegenerate case. To see the full boundary the point at infinity has to
be blown up in a nontrivial way. Such examples indicate that the situation
concerning the famous conjecture of Greene and Wu about existence of non
trivial bounded harmonic functions on CartanHadamard 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
