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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.

References [Enhancements On Off] (What's this?)

  • 1. W. Ballmann, M. Brin, K. Burns,
    On the differentiability of horocycles and horocycle foliations,
    J. Differential Geom. 26 (1987), 337-347. MR 0906395 (88k:53059)
  • 2. Y. Benoist, P. Foulon, F. Labourie,
    Flots d'Anosov à distributions stable et instable différentiables,
    J. Amer. Math. Soc. 5 (1992), 33-74. MR 1124979 (93b:58112)
  • 3. A. L. Besse,
    Manifolds all of whose Geodesics are Closed,
    Springer-Verlag, Berlin, Heidelberg, 1978. MR 0496885 (80c:53044)
  • 4. G. Besson, G. Courtois, S. Gallot,
    Entropies et rigidités des espaces localement symétriques de courbure strictement négative,
    Geom. Funct. Anal. 5 (1995), 731-799. MR 1354289 (96i:58136)
  • 5. M. Brin,
    Ergodic theory of frame flows.
    In: A. Katok (ed), Ergodic theory and dynamical systems, II (College Park, Md., 1979/1980), pp. 163-183, Progr. Math. 21, Birkhäuser, Boston, 1982. MR 0670078 (83m:58059)
  • 6. E. Damek, F. Ricci,
    A class of nonsymmetric harmonic Riemannian spaces,
    Bull. Amer. Math. Soc. 27 (1992), 139-142. MR 1142682 (93b:53043)
  • 7. J.-H. Eschenburg, E. Heintze,
    Comparison theory for Riccati equations,
    Manuscripta Math. 68 (1990), 209-214. MR 1063226 (91d:34034)
  • 8. P. Foulon, F. Labourie,
    Sur les variétés compactes asymptotiquement harmoniques,
    Invent. Math. 109 (1992), 97-111. MR 1168367 (93g:58114)
  • 9. J. Heber,
    On harmonic and asymptotically harmonic homogeneous spaces,
    To appear in Geom. Funct. Anal.
  • 10. E. Heintze, H. Im Hof,
    Geometry of horospheres,
    J. Differential Geom. 12 (1977), 481-491. MR 0512919 (80a:53051)
  • 11. G. Knieper,
    Spherical means on compact Riemannian manifolds of negative curvature,
    Differential Geom. Appl. 4 (1994), 361-390. MR 1306567 (95i:58141)
  • 12. G. Knieper,
    Hyperbolic Dynamics and Riemannian Geometry.
    In: B. Hasselblatt, A. Katok, Handbook of Dynamical Systems, Vol. 1A, pp. 453-545, North Holland, Amsterdam, 2002. MR 1928523 (2003i:37023)
  • 13. P. Petersen,
    Riemannian Geometry,
    Springer-Verlag, New York, 1998. MR 1480173 (98m:53001)
  • 14. A. Ranjan, H. Shah,
    Busemann functions in a harmonic manifold,
    Geom. Dedicata 101 (2003), 167-183. MR 2017901 (2004k:53052)
  • 15. T. J. Willmore,
    Riemannian Geometry,
    Oxford University Press, New York, 1993. MR 1261641 (95e:53002)

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Additional Information

Jens Heber
Affiliation: Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany

Gerhard Knieper
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany

Hemangi M. Shah
Affiliation: Department of Mathematics, Indian Institute of Technology, Powai, Mumbai 400076, India

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

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