Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Nayatani's metric and conformal transformations of a Kleinian manifold


Author: Yasuhiro Yabuki
Journal: Proc. Amer. Math. Soc. 136 (2008), 301-310
MSC (2000): Primary 53A30; Secondary 22E40
DOI: https://doi.org/10.1090/S0002-9939-07-09022-3
Published electronically: October 5, 2007
MathSciNet review: 2350417
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: According to Schoen and Yau (1988), an extensive class of conformally flat manifolds is realized as Kleinian manifolds. Nayatani (1997) constructed a metric on a Kleinian manifold $ M$ which is compatible with the canonical flat conformal structure. He showed that this metric $ g_N$ has a large symmetry if $ g_N$ is a complete metric. Under certain assumptions including the completeness of $ g_N$, the isometry group of $ (M,g_N)$ coincides with the conformal transformation group of $ M$. In this paper, we show that $ g_N$ may have a large symmetry even if $ g_N$ is not complete. In particular, every conformal transformation is an isometry when $ (M,g_N)$ corresponds to a geometrically finite Kleinian group.


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

  • 1. A. Beardon and B. Maskit, Limit sets of Kleinian groups and finite sided fundamental polyhedra, Acta Math. 132 (1974), 1-12. MR 0333164 (48:11489)
  • 2. B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal. 113 (1993), 245-317. MR 1218098 (94e:57016)
  • 3. H. Izeki and S. Nayatani, Canonical metric on the domain of discontinuity of a Kleinian group, Séminaire de théorie spectrale et géométrie GRENOBLE 16 (1998), 9-32. MR 1666506 (2000a:53018)
  • 4. A. Marden, The geometry of finitely generated Kleinian groups, Ann. of Math. (2) 99 (1974), 383-462. MR 0349992 (50:2485)
  • 5. J. Maubon, Geometrically finite Kleinian groups: The completeness of Nayatani's metric, C. R. Acad. Sci. Paris, 325 (1997), 1065-1070. MR 1614003 (99a:30041)
  • 6. S. Nayatani, Patterson-Sullivan measure and conformally flat metrics, Math. Z. 225 (1997), 115-131. MR 1451336 (98g:53072)
  • 7. S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-273. MR 0450547 (56:8841)
  • 8. P. J. Nicholls, The ergodic theory of discrete groups, LMS Lect. Notes Ser. 143, Cambridge University Press, Cambridge, 1989. MR 1041575 (91i:58104)
  • 9. R. Schoen and S. T. Yau, Conformally flat manifolds, Kleinian groups and scaler curvature, Invent. Math. 92, (1988) 47-71. MR 931204 (89c:58139)
  • 10. D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, I. H. E. S. Publ. Math. 50 (1979), 171-202. MR 556586 (81b:58031)
  • 11. D. Sullivan, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), 259-277. MR 766265 (86c:58093)
  • 12. W. P. Thurston, The Geometry and Topology of 3-manifolds, notes, Princeton Univ. Math. Department, 1979.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53A30, 22E40

Retrieve articles in all journals with MSC (2000): 53A30, 22E40


Additional Information

Yasuhiro Yabuki
Affiliation: Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan
Email: sa3m30@math.tohoku.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-07-09022-3
Keywords: Nayatani's metric, geometrically finite, conformally flat.
Received by editor(s): June 15, 2006
Received by editor(s) in revised form: November 24, 2006
Published electronically: October 5, 2007
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society