Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



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

Author: Jeffrey F. Brock
Journal: J. Amer. Math. Soc. 16 (2003), 495-535
MSC (2000): Primary 30F40; Secondary 30F60, 37F30
Published electronically: March 4, 2003
MathSciNet review: 1969203
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a coarse interpretation of the Weil-Petersson distance $d_{{WP}}(X,Y)$ between two finite area hyperbolic Riemann surfaces $X$ and $Y$ using a graph of pants decompositions introduced by Hatcher and Thurston. The combinatorics of the pants graph reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold $Q(X,Y)$ with $X$ and $Y$ in its conformal boundary is comparable to the Weil-Petersson distance $d_{{WP}}(X,Y)$. In applications, we relate the Weil-Petersson distance to the Hausdorff dimension of the limit set and the lowest eigenvalue of the Laplacian for $Q(X,Y)$, and give a new finiteness criterion for geometric limits.

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

  • [Ah] L. Ahlfors.
    An extension of Schwarz's lemma.
    Trans. Amer. Math. Soc. 43(1938), 359-364.
  • [BP] R. Benedetti and C. Petronio.
    Lectures on Hyperbolic Geometry.
    Springer-Verlag, 1992. MR 94e:57015
  • [Brs1] L. Bers.
    Simultaneous uniformization.
    Bull. AMS 66(1960), 94-97. MR 22:2694
  • [Brs2] L. Bers.
    On boundaries of Teichmüller spaces and on Kleinian groups: I.
    Annals of Math. 91(1970), 570-600. MR 45:7044
  • [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
  • [Bon] F. Bonahon.
    Bouts des variétés hyperboliques de dimension 3.
    Annals of Math. 124(1986), 71-158. MR 88c:57013
  • [Bow] R. Bowen.
    Hausdorff dimension of quasi-circles.
    Publ. Math. IHES 50(1979), 11-25. MR 81g:57023
  • [Br1] J. Brock.
    Continuity of Thurston's length function.
    Geom. & Funct. Anal. 10(2000), 741-797. MR 2001g:57028
  • [Br2] J. Brock.
    Iteration of mapping classes and limits of hyperbolic 3-manifolds.
    Invent. Math. 143(2001), 523-570. MR 2002d:30052
  • [Br3] J. Brock.
    Weil-Petersson translation distance and volumes of mapping tori.
    To appear, Comm. Anal. Geom.
  • [BF] J. Brock and B. Farb.
    Rank and curvature of Teichmüller space.
    Preprint (2001). Submitted for publication.
  • [BC] M. Burger and R. Canary.
    A lower bound on $\lambda_0$ for geometrically finite hyperbolic $n$-manifolds.
    J. Reine Angew. Math. 454(1994), 37-57. MR 95h:58138
  • [Bus] P. Buser.
    Geometry and spectra of compact Riemann surfaces.
    Birkhauser Boston, 1992. MR 93g:58149
  • [Can1] R. D. Canary.
    On the Laplacian and the geometry of hyperbolic 3-manifolds.
    J. Diff. Geom. 36(1992), 349-367. MR 93g:57016
  • [Can2] R. D. Canary.
    Ends of hyperbolic 3-manifolds.
    J. Amer. Math. Soc. 6(1993), 1-35. MR 93e:57019
  • [Can3] R. D. Canary.
    A covering theorem for hyperbolic 3-manifolds and its applications.
    Topology 35(1996), 751-778. MR 97e:57012
  • [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
  • [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
  • [Gard] F. Gardiner.
    Teichmüller theory and quadratic differentials.
    Wiley Interscience, 1987. MR 88m:32044
  • [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
  • [Hat] A. Hatcher.
    On triangulations of surfaces.
    Topology Appl. 40(1991), 189-194. MR 92f:57020
  • [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
  • [HT] A. Hatcher and W. Thurston.
    A presentation for the mapping class group.
    Topology 19(1980), 221-237. MR 81k:57008
  • [IT] Y. Imayoshi and M. Taniguchi.
    An introduction to Teichmüller spaces.
    Springer-Verlag, 1992. MR 94b:32031
  • [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
  • [MM1] H. Masur and Y. Minsky.
    Geometry of the complex of curves I: hyperbolicity.
    Invent. Math. 138(1999), 103-149. MR 2000i:57027
  • [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
  • [Mc1] C. McMullen.
    Iteration on Teichmüller space.
    Invent. math. 99(1990), 425-454. MR 91a:57008
  • [Mc2] C. McMullen.
    Renormalization and 3-manifolds which fiber over the circle.
    Annals of Math. Studies 142, Princeton University Press, 1996. MR 97f:57022
  • [Mc3] C. McMullen.
    Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups.
    J. Diff. Geom. 51(1999), 471-515. MR 2001c:37045
  • [Min] Y. Minsky.
    The classification of punctured torus groups.
    Annals of Math. 149(1999), 559-626. MR 2000f:30028
  • [Ot] J. P. Otal.
    Le théorème d'hyperbolisation pour les variétés fibrées de dimension trois.
    Astérisque, 1996. MR 97e:57013
  • [Sul1] D. Sullivan.
    Travaux de Thurston sur les groupes quasi-fuchsiens et sur les variétés hyperboliques de dimension 3 fibrées sur $S^1$.
    Sem. Bourbaki 554(1979/80). MR 83h:58079
  • [Sul2] D. Sullivan.
    Aspects of positivity in Riemannian geometry.
    J. Diff. Geom. 25(1987), 327-351. MR 88d:58132
  • [Ta] E. Taylor.
    Geometric finiteness and the convergence of Kleinian groups.
    Comm. Anal. Geom. 5(1997), 497-533. MR 98k:30064
  • [Th1] W. P. Thurston.
    Geometry and topology of three-manifolds.
    Princeton lecture notes, 1979.
  • [Th2] W. P. Thurston.
    Hyperbolic structures on 3-manifolds I: Deformations of acylindrical manifolds.
    Annals of Math. 124(1986), 203-246. MR 88g:57014
  • [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
  • [Wol1] S. Wolpert.
    Noncompleteness of the Weil-Petersson metric for Teichmüller space.
    Pacific J. Math. 61(1975), 573-577. MR 54:10678
  • [Wol2] S. Wolpert.
    Chern forms and the Riemann tensor for the moduli space of curves.
    Invent. Math. 85(1986), 119-145. MR 87j:32070
  • [Wol3] S. Wolpert.
    Geodesic length functions and the Nielsen problem.
    J. Diff. Geom. 25(1987), 275-296. MR 88e:32032
  • [Wol4] S. Wolpert.
    The hyperbolic metric and the geometry of the universal curve.
    J. Diff. Geom. 31(1990), 417-472. MR 91a:32030
  • [Wol5] S. Wolpert.
    The geometry of the Weil-Petersson completion of Teichmüller space.
    Preprint (2002).

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 30F40, 30F60, 37F30

Retrieve articles in all journals with MSC (2000): 30F40, 30F60, 37F30

Additional Information

Jeffrey F. Brock
Affiliation: Mathematics Department, University of Chicago, 5734 S. University Avenue, Chicago, Illinois 60637

Keywords: Hyperbolic manifold, Kleinian group, pants decomposition, Teichm\"uller space, Weil-Petersson metric, limit set
Received by editor(s): October 30, 2001
Published electronically: March 4, 2003
Additional Notes: Research partially supported by NSF grant DMS-0072133 and an NSF postdoctoral fellowship.
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society