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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotically harmonic spaces in dimension 3


Authors: Jens Heber, Gerhard Knieper and Hemangi M. Shah
Journal: Proc. Amer. Math. Soc. 135 (2007), 845-849
MSC (2000): Primary 53C35; Secondary 53C25
Published electronically: August 31, 2006
MathSciNet review: 2262881
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Abstract: Let $ M$ be a Hadamard manifold of dimension $ 3$ whose sectional curvature satisfies $ -b^2 \le K \le -{a^2}< 0$ and whose curvature tensor satisfies $ \Vert\nabla R\Vert\le C$ for suitable constants $ 0<a\le b$ and $ C\ge 0$. We show that $ M$ is of constant sectional curvature provided $ M$ is asymptotically harmonic. This was previously only known if $ M$ admits a compact quotient.


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Additional Information

Jens Heber
Affiliation: Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany
Email: heber@math.uni-kiel.de

Gerhard Knieper
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
Email: Gerhard.Knieper@rub.de

Hemangi M. Shah
Affiliation: Department of Mathematics, Indian Institute of Technology, Powai, Mumbai 400076, India
Email: hema@math.iitb.ac.in

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08520-0
PII: S 0002-9939(06)08520-0
Keywords: Asymptotically harmonic manifold, horospheres, asymptotic geodesics.
Received by editor(s): April 19, 2005
Received by editor(s) in revised form: October 3, 2005
Published electronically: August 31, 2006
Additional Notes: All three authors were supported in part by DFG priority program “Global Differential Geometry" (SPP 1154)
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2006 American Mathematical Society