Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives. (English) Zbl 1061.30037

With his hyperbolic Dehn surgery theorem and later the orbifold theorem, Thurston demonstrated the power of using hyperbolic cone-manifolds to understand complete, non-singular hyperbolic 3-manifolds. In this paper, the author uses Hodgson and Kerckhoff’s techniques to study infinite volume hyperbolic 3-manifolds. The obtained results have many applications: the Bers density conjecture, the density of cusps on the boundary of quasiconformal deformations spaces, and for constructing type preserving sequences of Kleinian groups. The simplest example of the studied problem is the following: Let \(M\) be a hyperbolic 3-manifold and \(c\) a simple closed geodesic in \(M\). Then the topological manifold \(M\setminus c\) also has a complete hyperbolic metric \(\widehat M\). How does the geometry of \(M\) relate to that of \(\widehat M\)? Another question is the following: Assume \(\Gamma\) is a Kleinian group and \(\Gamma_i\) is a sequence of geometrically finite Kleinian groups such that \(\Gamma_i\to\Gamma\), algebraically. Does there exist a type preserving sequence \(\Gamma_i'\) of geometrically finite groups also converging to \(\Gamma\)? Here type preserving means that if elements \(\gamma_i\) converge to \(\gamma\), then \(\gamma\) is parabolic if and only if the \(\gamma_i\) are parabolic. The starting point is the local parametrization of hyperbolic cone-manifolds. The author describes some of Hodgson and Kerckhoff’s results on tubes in cone-manifolds and estimates on the radius of embedded tubes about the cone-singularity, and uses these ideas to show that embedded hyperbolic half spaces are disjoint from these tubular neighborhoods of the cone-singularity. The analytic deformation theory of cone-manifolds is used to control the length of geodesies as the cone-angle decreases. The key result of the paper is the following: The \(L^2\)-norm of the cone-manifold deformation bounds the change in projective structure. First this is done for a hyperbolic half space and then a large embedded half space is found in each geometrically finite end which allows one to globally bound the deformation of the entire projective structure. The final step is to understand the geometric limit of a sequence of cone-manifolds. Given a geometrically finite cone-manifold, under certain conditions, there is a one-parameter family of cone-manifolds decreasing the cone-angle to zero.


30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30-06 Proceedings, conferences, collections, etc. pertaining to functions of a complex variable
57M50 General geometric structures on low-dimensional manifolds
Full Text: DOI arXiv


[1] C. G. Anderson. Projective structures on Riemann surfaces and developing maps to \(\mathbb{H} ^3 \) and \(\mathbb{C} P^n\). Preprint (1999).
[2] James W. Anderson and Richard D. Canary, Cores of hyperbolic 3-manifolds and limits of Kleinian groups, Amer. J. Math. 118 (1996), no. 4, 745 – 779. · Zbl 0863.30048
[3] James W. Anderson and Richard D. Canary, Cores of hyperbolic 3-manifolds and limits of Kleinian groups. II, J. London Math. Soc. (2) 61 (2000), no. 2, 489 – 505. · Zbl 0959.30028
[4] Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. · Zbl 0591.53001
[5] Lipman Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570 – 600. · Zbl 0197.06001
[6] J. Brock and K. Bromberg. On the density of geometrically finite Kleinian groups. To appear Acta Math. · Zbl 1055.57020
[7] Jeffrey Brock, Kenneth Bromberg, Richard Evans, and Juan Souto, Tameness on the boundary and Ahlfors’ measure conjecture, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 145 – 166. · Zbl 1060.30054
[8] K. Bromberg, Rigidity of geometrically finite hyperbolic cone-manifolds, Geom. Dedicata 105 (2004), 143 – 170. · Zbl 1057.53029
[9] K. Bromberg. Projective structures with degenerate holonomy and the Bers’ density conjecture. 2002 preprint available at front.math.ucdavis.edu/math.GT/0211402. · Zbl 1137.30014
[10] R. D. Canary, The conformal boundary and the boundary of the convex core, Duke Math. J. 106 (2001), no. 1, 193 – 207. · Zbl 1012.57021
[11] Richard D. Canary, Marc Culler, Sa’ar Hersonsky, and Peter B. Shalen, Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups, J. Differential Geom. 64 (2003), no. 1, 57 – 109. · Zbl 1069.57004
[12] R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3 – 92. · Zbl 0612.57009
[13] R. D. Canary and S. Hersonsky. Ubiquity of geometric finiteness in boundaries of deformation spaces of hyperbolic 3-manifolds. To appear Amer. J. of Math. · Zbl 1062.57021
[14] Richard D. Canary and Yair N. Minsky, On limits of tame hyperbolic 3-manifolds, J. Differential Geom. 43 (1996), no. 1, 1 – 41. · Zbl 0856.57011
[15] C. Epstein. Envelopes of horospheres and Weingarten surfaces in hyperbolic 3-space. preprint.
[16] R. Evans. Tameness persists. To appear Amer. J. Math.
[17] Craig D. Hodgson and Steven P. Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998), no. 1, 1 – 59. · Zbl 0919.57009
[18] Craig D. Hodgson and Steven P. Kerckhoff, Harmonic deformations of hyperbolic 3-manifolds, Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001) London Math. Soc. Lecture Note Ser., vol. 299, Cambridge Univ. Press, Cambridge, 2003, pp. 41 – 73. · Zbl 1051.57018
[19] C. Hodgson and S. Kerckhoff. The shape of hyperbolic Dehn surgery space. In preparation. · Zbl 1144.57015
[20] C. Hodgson and S. Kerckhoff. Universal bounds for hyperbolic Dehn surgery. 2002 preprint available at front.math.ucdavis.edu/math.GT/0204345. · Zbl 1087.57011
[21] Shigeru Kodani, Convergence theorem for Riemannian manifolds with boundary, Compositio Math. 75 (1990), no. 2, 171 – 192. · Zbl 0703.53043
[22] Sadayoshi Kojima, Deformations of hyperbolic 3-cone-manifolds, J. Differential Geom. 49 (1998), no. 3, 469 – 516. · Zbl 0990.57004
[23] Curt McMullen, Cusps are dense, Ann. of Math. (2) 133 (1991), no. 1, 217 – 247. · Zbl 0718.30033
[24] Jean-Pierre Otal, Les géodésiques fermées d’une variété hyperbolique en tant que nœuds, Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001) London Math. Soc. Lecture Note Ser., vol. 299, Cambridge Univ. Press, Cambridge, 2003, pp. 95 – 104 (French, with English and French summaries). · Zbl 1049.57007
[25] I. Rivin and J-M. Schlenker. On the Schläfli differential formula. 1998 preprint available at front.math.ucdavis.edu/math.DG/0001176.
[26] Hung Hsi Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), no. 2, i – xii and 289 – 538. · Zbl 0875.58014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.