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Nayatani's metric and conformal transformations of a Kleinian manifold
Author(s):
Yasuhiro
Yabuki
Journal:
Proc. Amer. Math. Soc.
136
(2008),
301-310.
MSC (2000):
Primary 53A30;
Secondary 22E40
Posted:
October 5, 2007
MathSciNet review:
2350417
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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  which is compatible with the canonical flat conformal structure. He showed that this metric  has a large symmetry if  is a complete metric. Under certain assumptions including the completeness of  , the isometry group of  coincides with the conformal transformation group of  . In this paper, we show that  may have a large symmetry even if  is not complete. In particular, every conformal transformation is an isometry when  corresponds to a geometrically finite Kleinian group.
References:
-
- 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.
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Additional Information:
Yasuhiro
Yabuki
Affiliation:
Mathematical Institute, Tohoku University, 980-8578 Sendai, Japan
Email:
sa3m30@math.tohoku.ac.jp
DOI:
10.1090/S0002-9939-07-09022-3
PII:
S 0002-9939(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
Posted:
October 5, 2007
Communicated by:
Richard A. Wentworth
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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