Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

Knieper, Gerhard; Peyerimhoff, Norbert

doi:10.1007/s12220-015-9570-1

Harmonic Functions on Rank One Asymptotically Harmonic Manifolds

Knieper, Gerhard; Peyerimhoff, Norbert

Authors

Gerhard Knieper

Professor Norbert Peyerimhoff norbert.peyerimhoff@durham.ac.uk
Professor

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
Related Public URLs	http://arxiv.org/pdf/1404.4290.pdf