Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

Knieper, Gerhard; Peyerimhoff, Norbert

Abstract

Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature hh. In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds XX with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon–Nikodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horospherical means of bounded eigenfunctions implying that these eigenfunctions do not admit non-trivial continuous extensions to the geometric compactification X¯¯¯¯X¯.

Citation

Knieper, G., & Peyerimhoff, N. (2015). Harmonic Functions on Rank One Asymptotically Harmonic Manifolds. Journal of Geometric Analysis, 26(2), 750-781. https://doi.org/10.1007/s12220-015-9570-1

• Journal Article Type: Article
• Acceptance Date: Dec 15, 2014
• Online Publication Date: Feb 3, 2015
• Publication Date: Feb 3, 2015
• Deposit Date: Jan 6, 2016
• Publicly Available Date: Mar 31, 2016
• Journal: Journal of Geometric Analysis
• Print ISSN: 1050-6926
• Electronic ISSN: 1559-002X
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 26
• Issue: 2
• Pages: 750-781
• DOI: https://doi.org/10.1007/s12220-015-9570-1
• Public URL: https://durham-repository.worktribe.com/output/1395326
• Related Public URLs: http://arxiv.org/pdf/1404.4290.pdf