The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores

Journal of the American Mathematical Society

Volumn 16, Issue 3, 2003, Pages 495-535

  Brock, Jeffrey F ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

Hyperbolic manifold; Kleinian group; Limit set; Pants decomposition; Teichm ller space; Weil Petersson metric

Indexed keywords

EID: 0038129414     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-03-00424-7     Document Type: Article

References (43)

1. [Ah] L. Ahlfors, An extension of Schwarz's lemma. Trans. Amer. Math. Soc. 43(1938), 359-364.
2. [BP] R. Benedetti and C. Petronio. Lectures on Hyperbolic Geometry. Springer-Verlag, 1992. MR 94e:57015
3. [Brs1] L. Bers. Simultaneous uniformization. Bull. AMS 66(1960), 94-97. MR 22:2694
4. [Brs2] L. Bers. On boundaries of Teichmüller spaces and on Kleinian groups: I. Annals of Math. 91(1970), 570-600. MR 45:7044
5. [Brs3] L. Bers. Spaces of degenerating Riemann surfaces. In Discontinuous groups and Riemann surfaces, pages 43-55. Annals of Math Studies 76, Princeton University Press, 1974. MR 50:13497
6. [Bon] F. Bonahon. Bouts des variétés hyperboliques de dimension 3. Annals of Math. 124(1986), 71-158. MR 88c:57013
7. [Bow] R. Bowen. Hausdorff dimension of quasi-circles. Publ. Math. IHES 50(1979), 11-25. MR 81g:57023
8. [Br1] J. Brock. Continuity of Thurston's length function. Geom. & Funct. Anal. 10(2000), 741-797. MR 2001g:57028
9. [Br2] J. Brock. Iteration of mapping classes and limits of hyperbolic 3-manifolds. Invent. Math. 143(2001), 523-570. MR 2002d:30052
10. [Br3] J. Brock. Weil-Petersson translation distance and volumes of mapping tori. To appear, Comm. Anal. Geom.
11. [BF] J. Brock and B. Farb. Rank and curvature of Teichmüller space. Preprint (2001). Submitted for publication.
12. 0 for geometrically finite hyperbolic n-manifolds. J. Reine Angew. Math. 454(1994), 37-57. MR 95h:58138
13. [Bus] P. Buser. Geometry and spectra of compact Riemann surfaces. Birkhauser Boston, 1992. MR 93g:58149
14. [Can1] R. D. Canary. On the Laplacian and the geometry of hyperbolic 3-manifolds. J. Diff. Geom. 36(1992), 349-367. MR 93g:57016
15. [Can2] R. D. Canary. Ends of hyperbolic 3-manifolds. J. Amer. Math. Soc. 6(1993), 1-35. MR 93e:57019
16. [Can3] R. D. Canary. A covering theorem for hyperbolic 3-manifolds and its applications. Topology 35(1996), 751-778. MR 97e:57012
17. [CM] R. D. Canary and Y. N. Minsky. On limits of tame hyperbolic 3-manifolds. J. Diff. Geom. 43(1996), 1-41. MR 98f:57021
18. [EM] D. B. A. Epstein and A. Marden. Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces. In Analytical and Geometric Aspects of Hyperbolic Space, pages 113-254. Cambridge University Press, 1987. MR 89c:52014
19. [Gard] F. Gardiner. Teichmüller theory and quadratic differentials. Wiley Interscience, 1987. MR 88m:32044
20. [Har] W. J. Harvey. Boundary structure of the modular group. In Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference. Annals of Math Studies 97, Princeton University Press, 1981. MR 83d:32022
21. [Hat] A. Hatcher. On triangulations of surfaces. Topology Appl. 40(1991), 189-194. MR 92f:57020
22. [HLS] A. Hatcher, P. Lochak, and L. Schneps. On the Teichmüller tower of mapping class groups. J. Reine Angew. Math. 521(2000), 1-24. MR 2001h:57018
23. [HT] A. Hatcher and W. Thurston. A presentation for the mapping class group. Topology 19(1980), 221-237. MR 81k:57008
24. [IT] Y. Imayoshi and M. Taniguchi. An introduction to Teichmu̧ller spaces. Springer-Verlag, 1992. MR 94b:32031
25. [Mas] H. Masur. The extension of the Weil-Petersson metric to the boundary of Teichmüller space. Duke Math. J. 43(1976), 623-635. MR 54:5506
26. [MM1] H. Masur and Y. Minsky. Geometry of the complex of curves I: hyperbolicity. Invent. Math. 138(1999), 103-149. MR 20001:57027
27. [MM2] H. Masur and Y. Minsky. Geometry of the complex of curves II: hierarchical structure. Geom. & Funct. Anal. 10(2000), 902-974. MR 2001k:57020
28. [MC1] C. McMullen. Iteration on Teichmüller space. Invent, math. 99(1990), 425-454. MR 91a:57008
29. [MC2] C. McMullen. Renormalization and 3-manifolds which fiber over the circle. Annals of Math. Studies 142, Princeton University Press, 1996. MR 97f:57022
30. [MC3] C. McMullen. Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups. J. Diff. Geom. 51(1999), 471-515. MR 2001c:37045
31. [Min] Y. Minsky. The classification of punctured torus groups. Annals of Math. 149(1999), 559-626. MR 2000f:30028
32. [Ot] J. P. Otal. Le théorème d'hyperbolisation pour les variétés fibrées de dimension trois. Astérisque, 1996. MR 97e:57013
33. 1. Sem. Bourbaki 554(1979/80). MR 83h:58079
34. [Sul2] D. Sullivan. Aspects of positivity in Riemannian geometry. J. Diff. Geom. 25(1987), 327-351. MR 88d:58132
35. [Ta] E. Taylor. Geometric finiteness and the convergence of Kleinian groups. Comm. Anal. Geom. 5(1997), 497-533. MR 98k:30064
36. [Th1] W. P. Thurston. Geometry and topology of three-manifolds. Princeton lecture notes, 1979.
37. [Th2] W. P. Thurston. Hyperbolic structures on 3-manifolds I: Deformations of acylindrical manifolds. Annals of Math. 124(1986), 203-246. MR 88g:57014
38. [Tro] A. J. Tromba. On a natural algebraic affine connection on the space of almost complex structures and the curvature of Teichmüller space with respect to its Weil-Petersson metric. Manuscripta Math. 56(1986), 475-497. MR 88c:32034
39. [Wol1] S. Wolpert. Noncompleteness of the Weil-Petersson metric for Teichmüller space. Pacific J. Math. 61(1975), 573-577. MR 54:10678
40. [Wol2] S. Wolpert. Chern forms and the Riemann tensor for the moduli space of curves. Invent. Math. 85(1986), 119-145. MR 87j:32070
41. [Wol3] S. Wolpert. Geodesic length functions and the Nielsen problem. J. Diff. Geom. 25(1987), 275-296. MR 88e:32032
42. [Wol4] S. Wolpert. The hyperbolic metric and the geometry of the universal curve. J. Diff. Geom. 31(1990), 417-472. MR 91a:32030
43. [Wol5] S. Wolpert. The geometry of the Weil-Petersson completion of Teichmüller space. Preprint (2002).