Complex earthquakes and Teichmüller theory

McMullen, Curtis

doi:10.1090/S0894-0347-98-00259-8

Complex earthquakes and Teichmüller theory
HTML articles powered by AMS MathViewer

by Curtis T. McMullen

J. Amer. Math. Soc. 11 (1998), 283-320

DOI: https://doi.org/10.1090/S0894-0347-98-00259-8

PDF | Request permission

Abstract:

It is known that any two points in Teichmüller space are joined by an earthquake path. In this paper we show any earthquake path $\mathbb R \rightarrow T(S)$ extends to a proper holomorphic mapping of a simply-connected domain $D$ into Teichmüller space, where $\mathbb R \subset D \subset \mathbb C$. These complex earthquakes relate Weil-Petersson geometry, projective structures, pleated surfaces and quasifuchsian groups. Using complex earthquakes, we prove grafting is a homeomorphism for all 1-dimensional Teichmüller spaces, and we construct bending coordinates on Bers slices and their generalizations. In the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold.

References

James W. Anderson and Richard D. Canary, Algebraic limits of Kleinian groups which rearrange the pages of a book, Invent. Math. 126 (1996), no. 2, 205–214. MR 1411128, DOI 10.1007/s002220050094
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, Holomorphic families of isomorphisms of Möbius groups, J. Math. Kyoto Univ. 26 (1986), no. 1, 73–76. MR 827159, DOI 10.1215/kjm/1250520965
Lipman Bers and Leon Ehrenpreis, Holomorphic convexity of Teichmüller spaces, Bull. Amer. Math. Soc. 70 (1964), 761–764. MR 168800, DOI 10.1090/S0002-9904-1964-11230-1
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
Francis Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), no. 1, 139–162. MR 931208, DOI 10.1007/BF01393996
Clifford J. Earle, On variation of projective structures, 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., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 87–99. MR 624807
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
Gerd Faltings, Real projective structures on Riemann surfaces, Compositio Math. 48 (1983), no. 2, 223–269. MR 700005
A. Fathi, F. Laudenbach, and V. Poénaru. Travaux de Thurston sur les surfaces. Astérisque, volume 66-67, 1979.
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
William M. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), no. 3, 297–326. MR 882826
J. J. Gray, Fuchs and the theory of differential equations, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 1, 1–26. MR 722855, DOI 10.1090/S0273-0979-1984-15186-3
R. C. Gunning, Lectures on vector bundles over Riemann surfaces, University of Tokyo Press, Tokyo; Princeton University Press, Princeton, N.J., 1967. MR 0230326
R. C. Gunning, Affine and projective structures on Riemann surfaces, 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., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 225–244. MR 624816
Dennis A. Hejhal, Monodromy groups and linearly polymorphic functions, Acta Math. 135 (1975), no. 1, 1–55. MR 463429, DOI 10.1007/BF02392015
C. Hodgson and S. Kerckhoff. Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery. Preprint, 1995.
John H. Hubbard, The monodromy of projective structures, 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., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 257–275. MR 624819
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
Yoshinobu Kamishima and Ser P. Tan, Deformation spaces on geometric structures, Aspects of low-dimensional manifolds, Adv. Stud. Pure Math., vol. 20, Kinokuniya, Tokyo, 1992, pp. 263–299. MR 1208313, DOI 10.2969/aspm/02010263
Linda Keen and Caroline Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), no. 4, 719–749. MR 1241870, DOI 10.1016/0040-9383(93)90048-Z
Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, DOI 10.2307/2007076
Steven P. Kerckhoff, Earthquakes are analytic, Comment. Math. Helv. 60 (1985), no. 1, 17–30. MR 787659, DOI 10.1007/BF02567397
Steven P. Kerckhoff and William P. Thurston, Noncontinuity of the action of the modular group at Bers’ boundary of Teichmüller space, Invent. Math. 100 (1990), no. 1, 25–47. MR 1037141, DOI 10.1007/BF01231179
Irwin Kra, Deformation spaces, A crash course on Kleinian groups (Lectures, Special Session, Annual Winter Meeting, Amer. Math. Soc., San Francisco, Calif., 1974) Lecture Notes in Math., Vol. 400, Springer, Berlin, 1974, pp. 48–70. MR 0402122
Olli Lehto, Univalent functions and Teichmüller spaces, Graduate Texts in Mathematics, vol. 109, Springer-Verlag, New York, 1987. MR 867407, DOI 10.1007/978-1-4613-8652-0
Bernard Maskit, On a class of Kleinian groups, Ann. Acad. Sci. Fenn. Ser. A I 442 (1969), 8. MR 0252638
Shigenori Matsumoto, Foundations of flat conformal structure, Aspects of low-dimensional manifolds, Adv. Stud. Pure Math., vol. 20, Kinokuniya, Tokyo, 1992, pp. 167–261. MR 1208312, DOI 10.2969/aspm/02010167
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
Y. Minsky. The classification of punctured torus groups. SUNY Preprint 1997/6.
D. Mumford, C. McMullen, and D. Wright. Limit sets of free two-generator kleinian groups. Preprint, 1990.
Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
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
John R. Parker and Caroline Series, Bending formulae for convex hull boundaries, J. Anal. Math. 67 (1995), 165–198. MR 1383492, DOI 10.1007/BF02787788
Hiroshige Shiga, On analytic and geometric properties of Teichmüller spaces, J. Math. Kyoto Univ. 24 (1984), no. 3, 441–452. MR 766636, DOI 10.1215/kjm/1250521274
H. Shiga and H. Tanigawa. Projective structures with discrete holonomy representations. Trans. Amer. Math. Soc., To appear.
Dennis Sullivan, Quasiconformal homeomorphisms and dynamics. II. Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), no. 3-4, 243–260. MR 806415, DOI 10.1007/BF02392543
H. Tanigawa. Grafting, harmonic maps and projective structures on surfaces. Preprint, 1995.
W. P. Thurston. Geometry and Topology of Three-Manifolds. Lecture Notes, Princeton University, 1979.
William P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 91–112. MR 903860
William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975, DOI 10.1515/9781400865321
W. P. Thurston. Hyperbolic structures on 3-manifolds II: Surface groups and 3-manifolds which fiber over the circle. Preprint.
Michael Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479. MR 982185
Scott Wolpert, An elementary formula for the Fenchel-Nielsen twist, Comment. Math. Helv. 56 (1981), no. 1, 132–135. MR 615620, DOI 10.1007/BF02566203
Scott Wolpert, The Fenchel-Nielsen deformation, Ann. of Math. (2) 115 (1982), no. 3, 501–528. MR 657237, DOI 10.2307/2007011
Scott Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), no. 4, 969–997. MR 796909, DOI 10.2307/2374363
Scott A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275–296. MR 880186
D. Wright. The shape of the boundary of Maskit’s embedding of the Teichmüller space of once-punctured tori. Preprint, 1990.