Abstract
Let (M, g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that if M is asymptotically harmonic of constant h = 0, then M is a flat manifold. This theorem shows that any asymptotically harmonic manifold in dimension 3 is a symmetric space, thus completing the classification of asymptotically harmonic manifolds in dimension 3.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Besson G., Courtois G., Gallot S.: Entropies et rigidités des espaces localement symétriques de courbure strictement négative. Geom. Funct. Anal. 5, 731–799 (1995)
Heber J., Knieper G., Shah H.: Asymptotically harmonic spaces in dimension 3. Proc. Amer. Math. Soc. 135, 845–849 (2007)
Knieper G.: Spherical means on compact Riemannian manifolds of negative curvature. Differential Geom. Appl. 4, 361–390 (1994)
Schroeder V., Shah H.: On 3-dimensional asymptotically harmonic manifolds. Arch. Math. 90, 275–278 (2008)
A. Zimmer, Asymptotically harmonic manifolds without focal points, arXiv:1109.2481v2 [math.DG] 6 Oct 2011
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shah, H. 3-dimensional asymptotically harmonic manifolds with minimal horospheres. Arch. Math. 106, 81–84 (2016). https://doi.org/10.1007/s00013-015-0837-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-015-0837-3