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Harmonic functions on rank one asymptotically harmonic manifolds.

Knieper, Gerhard and Peyerimhoff, Norbert (2015) 'Harmonic functions on rank one asymptotically harmonic manifolds.', The journal of geometric analysis., 26 (2). pp. 750-781.


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¯.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Publisher statement:The final publication is available at Springer via
Date accepted:15 December 2014
Date deposited:31 March 2016
Date of first online publication:03 February 2015
Date first made open access:No date available

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