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

Brock, Jeffrey

doi:10.1090/S0894-0347-03-00424-7

The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores
HTML articles powered by AMS MathViewer

by Jeffrey F. Brock;

J. Amer. Math. Soc. 16 (2003), 495-535

DOI: https://doi.org/10.1090/S0894-0347-03-00424-7

Published electronically: March 4, 2003

PDF | Request permission

Abstract:

We present a coarse interpretation of the Weil-Petersson distance $d_{\mathrm {WP}}(X,Y)$ 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 $d_{\mathrm {WP}}(X,Y)$. 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

Trans. Amer. Math. Soc.

43

Riccardo Benedetti and Carlo Petronio, Lectures on hyperbolic geometry, Universitext, Springer-Verlag, Berlin, 1992. MR 1219310, DOI 10.1007/978-3-642-58158-8
Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97. MR 111834, DOI 10.1090/S0002-9904-1960-10413-2
Lipman Bers, On boundaries of Teichmüller spaces and on Kleinian groups. I, Ann. of Math. (2) 91 (1970), 570–600. MR 297992, DOI 10.2307/1970638
Lipman Bers, Spaces of degenerating Riemann surfaces, Discontinuous groups and Riemann surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973) Ann. of Math. Stud., No. 79, Princeton Univ. Press, Princeton, NJ, 1974, pp. 43–55. MR 361051
Francis Bonahon, Bouts des variétés hyperboliques de dimension $3$, Ann. of Math. (2) 124 (1986), no. 1, 71–158 (French). MR 847953, DOI 10.2307/1971388
Rufus Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25. MR 556580, DOI 10.1007/BF02684767
J. F. Brock, Continuity of Thurston’s length function, Geom. Funct. Anal. 10 (2000), no. 4, 741–797. MR 1791139, DOI 10.1007/PL00001637
Jeffrey F. Brock, Iteration of mapping classes and limits of hyperbolic 3-manifolds, Invent. Math. 143 (2001), no. 3, 523–570. MR 1817644, DOI 10.1007/PL00005799

To appear, Comm. Anal. Geom.

Preprint (2001). Submitted for publication.

Marc Burger and Richard D. Canary, A lower bound on $\lambda _0$ for geometrically finite hyperbolic $n$-manifolds, J. Reine Angew. Math. 454 (1994), 37–57. MR 1288678
Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
Richard D. Canary, On the Laplacian and the geometry of hyperbolic $3$-manifolds, J. Differential Geom. 36 (1992), no. 2, 349–367. MR 1180387
Richard D. Canary, Ends of hyperbolic $3$-manifolds, J. Amer. Math. Soc. 6 (1993), no. 1, 1–35. MR 1166330, DOI 10.1090/S0894-0347-1993-1166330-8
Richard D. Canary, A covering theorem for hyperbolic $3$-manifolds and its applications, Topology 35 (1996), no. 3, 751–778. MR 1396777, DOI 10.1016/0040-9383(94)00055-7
Richard D. Canary and Yair N. Minsky, On limits of tame hyperbolic $3$-manifolds, J. Differential Geom. 43 (1996), no. 1, 1–41. MR 1424418
D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852
Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 903027
W. J. Harvey, Boundary structure of the modular group, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., No. 97, Princeton Univ. Press, Princeton, NJ, 1981, pp. 245–251. MR 624817
Allen Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991), no. 2, 189–194. MR 1123262, DOI 10.1016/0166-8641(91)90050-V
Allen Hatcher, Pierre Lochak, and Leila Schneps, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521 (2000), 1–24. MR 1752293, DOI 10.1515/crll.2000.028
A. Hatcher and W. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), no. 3, 221–237. MR 579573, DOI 10.1016/0040-9383(80)90009-9
Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338, DOI 10.1007/s002220050343
H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000), no. 4, 902–974. MR 1791145, DOI 10.1007/PL00001643
C. McMullen, Iteration on Teichmüller space, Invent. Math. 99 (1990), no. 2, 425–454. MR 1031909, DOI 10.1007/BF01234427
Curtis T. McMullen, Renormalization and 3-manifolds which fiber over the circle, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR 1401347, DOI 10.1515/9781400865178
Curtis T. McMullen, Hausdorff dimension and conformal dynamics. I. Strong convergence of Kleinian groups, J. Differential Geom. 51 (1999), no. 3, 471–515. MR 1726737
Yair N. Minsky, The classification of punctured-torus groups, Ann. of Math. (2) 149 (1999), no. 2, 559–626. MR 1689341, DOI 10.2307/120976
Jean-Pierre Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996), x+159 (French, with French summary). MR 1402300
Dennis Sullivan, Travaux de Thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension $3$ fibrées sur $S^{1}$, Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin, 1981, pp. 196–214 (French). MR 636524
Dennis Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), no. 3, 327–351. MR 882827
Edward C. Taylor, Geometric finiteness and the convergence of Kleinian groups, Comm. Anal. Geom. 5 (1997), no. 3, 497–533. MR 1487726, DOI 10.4310/CAG.1997.v5.n3.a5

Geometry and topology of three-manifolds

William P. Thurston, Hyperbolic structures on $3$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203–246. MR 855294, DOI 10.2307/1971277
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), no. 4, 475–497. MR 860734, DOI 10.1007/BF01168506
Scott Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller space, Pacific J. Math. 61 (1975), no. 2, 573–577. MR 422692, DOI 10.2140/pjm.1975.61.573
Scott A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119–145. MR 842050, DOI 10.1007/BF01388794
Scott A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275–296. MR 880186
Scott A. Wolpert, The hyperbolic metric and the geometry of the universal curve, J. Differential Geom. 31 (1990), no. 2, 417–472. MR 1037410

Preprint (2002).