2008/02/07 by Marc Arnaudon, Arnaudon, Marc, Anton Thalmaier +3
Mathematics · #58J65 (Primary) 60H30 (Secondary) #Advanced Harmonic Analysis Research #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR)
paper · doi:10.48550/arxiv.0802.0966
openalex publication_date 2008/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.