Hyperbolic conemanifolds, short geodesics, and Schwarzian derivatives
Author:
K. Bromberg
Journal:
J. Amer. Math. Soc. 17 (2004), 783826
MSC (2000):
Primary 30F40, 57M50
Published electronically:
July 21, 2004
MathSciNet review:
2083468
Fulltext PDF Free Access
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Abstract: Given a geometrically finite hyperbolic conemanifold, with the conesingularity sufficiently short, we construct a oneparameter family of conemanifolds decreasing the coneangle to zero. We also control the geometry of this oneparameter family via the Schwarzian derivative of the projective boundary and the length of closed geodesics.
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 [And]
 C. G. Anderson.
Projective structures on Riemann surfaces and developing maps to and . Preprint (1999).
 [AC1]
 J. Anderson and R. Canary.
Cores of hyperbolic 3manifolds and limits of Kleinian groups. Amer. J. Math. 118(1996), 745779. MR 1400058 (97k:57015)
 [AC2]
 J. Anderson and R. Canary.
Cores of hyperbolic 3manifolds and limits of Kleinian groups II. J. LMS 61(2000), 489505. MR 1760675 (2001h:30041)
 [BGS]
 W. Ballman, M. Gromov, and V. Schroeder.
Manifolds of Nonpositive Curvature. Birkhauser, 1985. MR 0823981 (87h:53050)
 [Bers]
 L. Bers.
On boundaries of Teichmüller spaces and on Kleinian groups: I. Annals of Math. 91(1970), 570600. MR 0297992 (45:7044)
 [BB]
 J. Brock and K. Bromberg.
On the density of geometrically finite Kleinian groups. To appear Acta Math.
 [BBES]
 J. Brock, K. Bromberg, R. Evans, and J. Souto.
Boundaries of deformation spaces and Ahlfors' measure conjecture. Publ. Math. I.H.É.S. 98(2003), 145166. MR 2031201
 [Br1]
 K. Bromberg.
Rigidity of geometrically finite hyperbolic conemanifolds. Geom. Dedicata 105(2004), 143170. MR 2057249
 [Br2]
 K. Bromberg.
Projective structures with degenerate holonomy and the Bers' density conjecture. 2002 preprint available at front.math.ucdavis.edu/math.GT/0211402.
 [Can]
 R. D. Canary.
The conformal boundary and the boundary of the convex core. Duke Math. J. 106(2000), 193207. MR 1810370 (2001m:57024)
 [CCHS]
 R. D. Canary, M. Culler, S. Hersonsky, and P. B. Shalen.
Approximation by maximal cusps in the boundaries of quasiconformal deformation spaces. J. Diff. Geom. 64(2003), 57109. MR 2015044
 [CEG]
 R. D. Canary, D. B. A. Epstein, and P. Green.
Notes on notes of Thurston. In Analytical and Geometric Aspects of Hyperbolic Space, pages 392. Cambridge University Press, 1987. MR 0903850 (89e:57008)
 [CH]
 R. D. Canary and S. Hersonsky.
Ubiquity of geometric finiteness in boundaries of deformation spaces of hyperbolic 3manifolds. To appear Amer. J. of Math.
 [CM]
 R. D. Canary and Y. N. Minsky.
On limits of tame hyperbolic 3manifolds. J. Diff. Geom. 43(1996), 141. MR 1424418 (98f:57021)
 [Ep]
 C. Epstein.
Envelopes of horospheres and Weingarten surfaces in hyperbolic 3space. preprint.
 [Ev]
 R. Evans.
Tameness persists. To appear Amer. J. Math.
 [HK1]
 C. Hodgson and S. Kerckhoff.
Rigidity of hyperbolic conemanifolds and hyperbolic Dehn surgery. J. Diff. Geom. 48(1998), 159. MR 1622600 (99b:57030)
 [HK2]
 C. Hodgson and S. Kerckhoff.
Harmonic deformations of hyperbolic 3manifolds. In Kleinian Groups and Hyperbolic 3manifolds. London Math Society Lecture Notes, Cambridge University Press, 2003. MR 2044544
 [HK3]
 C. Hodgson and S. Kerckhoff.
The shape of hyperbolic Dehn surgery space. In preparation.
 [HK4]
 C. Hodgson and S. Kerckhoff.
Universal bounds for hyperbolic Dehn surgery. 2002 preprint available at front.math.ucdavis.edu/math.GT/0204345.
 [Ko1]
 S. Kodani.
Convergence theorem for Riemannian manifolds with boundary. Compositio Math. 75(1990), 171192. MR 1065204 (92b:53066)
 [Ko2]
 S. Kojima.
Deformations of hyperbolic 3cone manifolds. J. Diff. Geom. 49(1998), 469516. MR 1669649 (2000d:57023)
 [Mc]
 C. McMullen.
Cusps are dense. Annals of Math. 133(1991), 217247. MR 1087348 (91m:30058)
 [Ot]
 J. P. Otal.
Les géodésiques fermées d'une variété hyperbolique en tant que noeuds. In Kleinian Groups and Hyperbolic 3manifolds. London Math Society Lecture Notes, Cambridge University Press, 2003. MR 2044546
 [RS]
 I. Rivin and JM. Schlenker.
On the Schläfli differential formula. 1998 preprint available at front.math.ucdavis.edu/math.DG/0001176.
 [Wu]
 H. Wu.
The Bochner technique in differential geometry. Math. Reports, London (1987). MR 1079031 (91h:58031)
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Additional Information
K. Bromberg
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, California 91125
Address at time of publication:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Email:
bromberg@math.utah.edu
DOI:
http://dx.doi.org/10.1090/S089403470400462X
PII:
S 08940347(04)00462X
Keywords:
Kleinian groups,
conemanifolds,
Schwarzian derivative
Received by editor(s):
December 10, 2002
Published electronically:
July 21, 2004
Additional Notes:
Supported by a grant from the NSF
Article copyright:
© Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
