Arnaudon, Marc and Thalmaier, Anton and Ulsamer, Stefanie (2009) Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature. MATHEMATISCHE ZEITSCHRIFT, 263 (2). pp. 369-409. ISSN 0025-5874, 1432-1823
Full text not available from this repository. (Request a copy)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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | NEGATIVELY CURVED MANIFOLDS; DIRICHLET PROBLEM; BROWNIAN-MOTION; LIMITING ANGLE; INFINITY; Harmonic function; Poisson boundary; Cartan-Hadamard manifold; Conjecture of Greene-Wu; Dirichlet problem at infinity |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 04 Sep 2020 07:10 |
| Last Modified: | 04 Sep 2020 07:10 |
| URI: | https://pred.uni-regensburg.de/id/eprint/28362 |
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