Duke Mathematical Journal

A symplectic map between hyperbolic and complex Teichmüller theory

Kirill Krasnov and Jean-Marc Schlenker

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let S be a closed, orientable surface of genus at least 2. The space TH×ML, where TH is the “hyperbolic” Teichmüller space of S and ML is the space of measured geodesic laminations on S, is naturally a real symplectic manifold. The space CP of complex projective structures on S is a complex symplectic manifold. A relation between these spaces is provided by Thurston's grafting map Gr. We prove that this map, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends

Article information

Source
Duke Math. J., Volume 150, Number 2 (2009), 331-356.

Dates
First available in Project Euclid: 16 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1255699343

Digital Object Identifier
doi:10.1215/00127094-2009-054

Mathematical Reviews number (MathSciNet)
MR2569616

Zentralblatt MATH identifier
1206.30058

Subjects
Primary: 30F60: Teichmüller theory [See also 32G15]
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

Citation

Krasnov, Kirill; Schlenker, Jean-Marc. A symplectic map between hyperbolic and complex Teichmüller theory. Duke Math. J. 150 (2009), no. 2, 331--356. doi:10.1215/00127094-2009-054. https://projecteuclid.org/euclid.dmj/1255699343


Export citation

References

  • L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand Math. Stud. 10, Van Nostrand, Toronto, 1966.
  • F. Bonahon, Shearing hyperbolic surfaces, bending pleated surfaces and Thurston's symplectic form, Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), 233--297.
  • —, Geodesic laminations with transverse Hölder distributions, Ann. Sci. École Norm. Sup. (4) 30 (1997), 205--240.
  • —, A Schläfli-type formula for convex cores of hyperbolic $3$-manifolds, J. Differential Geom. 50 (1998), 25--58.
  • —, Variations of the boundary geometry of, $3$-dimensional hyperbolic convex cores, J. Differential Geom. 50 (1998), 1--24.
  • F. Bonsante and J.-M. Schlenker, AdS manifolds with particles and earthquakes on singular surfaces, Geom. Funct. Anal. 19 (2009), 41--82.
  • D. Dumas, ``Complex projective structures'' to appear in Handbook of Teichmüller Theory, Vol. 2, IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich, 2009, 455--508.; preprint,\arxiv0902.1951v1[math.DG]
  • D. Dumas and M. Wolf, Projective structures, grafting and measured laminations, Geom. Topol. 12 (2008), 351--386.
  • C. L. Epstein, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math. 372 (1986), 96--135.
  • —, Envelopes of horospheres and Weingarten surfaces in hyperbolic $3$-space, preprint, 1984.
  • D. B. A. Epstein and A. Marden, ``Convex hulls in hyperbolic spaces, a theorem of Sullivan, and measured pleated surfaces'' in Analytical and Geometric Aspects of Hyperbolic Space, London Math. Soc. Lecture Note Ser. 111, Cambridge Univ. Press, Cambridge, 1986.
  • A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces, Séminaire Orsay, reprint of Travaux de Thurston sur les surfaces, Soc. Math. France, Montrouge, 1979, Astérisque 66--67 (1991).
  • W. Fenchel and J. Nielsen, Discontinuous Groups of Isometries in the Hyperbolic Plane, de Gruyter Stud. Math. 29, de Gruyter, Berlin, 2003.
  • W. M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), 200--225.
  • S. Kawai, The symplectic nature of the space of projective connections on Riemann surfaces, Math. Ann. 305 (1996), 161--182.
  • S. P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), 235--265.
  • —, personal communication.
  • K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000), 929--979.
  • K. Krasnov and J.-M. Schlenker, Minimal surfaces and particles in $3$-manifolds, Geom. Dedicata 126 (2007), 187--254.
  • —, On the renormalized volume of hyperbolic $3$-manifolds, Comm. Math. Phys. 279 (2008), 637--668.
  • C. T. Mcmullen, The moduli space of Riemann surfaces is Kähler hyperbolic, Ann. of Math. (2) 151 (2000), 327--357.
  • J. Milnor, Collected Papers, Vol. 1, Publish or Perish, Houston, 1994.
  • S. J. Patterson and P. A. Perry, The divisor of Selberg's zeta function for Kleinian groups, with an appendix by C. Epstein, Duke Math. J. 106 (2001), 321--390.
  • H. Poincaré, Sur l'uniformisation des fonctions analytiques, Acta Math. 31 (1908), 1--63.
  • I. Rivin and J.-M. Schlenker, The Schläfli formula in Einstein manifolds with boundary, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 18--23.
  • —, On the Schläfli differential formula, preprint,\arxivmath/0001176v2[math.DG]
  • K. P. Scannell and M. Wolf, The grafting map of Teichmüller space, J. Amer. Math. Soc. 15 (2002), 893--927.
  • J.-M. Schlenker, Hypersurfaces in $H\sp n$ and the space of its horospheres, Geom. Funct. Anal. 12 (2002), 395--435.
  • Y. SöZen and F. Bonahon, The Weil-Petersson and Thurston symplectic forms, Duke Math. J. 108 (2001), 581--597.
  • L. A. Takhtajan and L.-P. Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Comm. Math. Phys. 239 (2003), 183--240.
  • O. TeichmüLler, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 1939 (1940).
  • W. P. Thurston, Three-Dimensional Geometry and Topology, Vol. 1, Princeton Math. Ser. 35, Princeton Univ. Press, Princeton, 1997.
  • S. A. Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) 117 (1983), 207--234.
  • —, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), 969--997.
  • —, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), 275--296.
  • P. G. Zograf and L. A. Takhtajan, On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichmüller and Schottky spaces (in Russian), Mat. Sb. N.S. 132 (174), no. 3 (1987), 303--320.; English translation in Math. USSR Sb. 60 (1988), 297--313.