Geometry & Topology

A cyclic extension of the earthquake flow I

Francesco Bonsante, Gabriele Mondello, and Jean-Marc Schlenker

Full-text: Open access

Abstract

Let T be Teichmüller space of a closed surface of genus at least 2. We describe an action of the circle on T×T, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of T. This circle action shares some of the main properties of the earthquake flow, for instance it satisfies an extension of Thurston’s Earthquake Theorem and it has a complex extension which is analogous and limits to complex earthquakes. Moreover, a related circle action on T×T extends to the product of two copies of the universal Teichmüller space.

Article information

Source
Geom. Topol., Volume 17, Number 1 (2013), 157-234.

Dates
Received: 6 September 2011
Revised: 5 July 2012
Accepted: 13 August 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732522

Digital Object Identifier
doi:10.2140/gt.2013.17.157

Mathematical Reviews number (MathSciNet)
MR3035326

Zentralblatt MATH identifier
1278.57024

Subjects
Primary: 57M50: Geometric structures on low-dimensional manifolds

Keywords
anti-de Sitter earthquakes constant curvature surfaces space-like surfaces

Citation

Bonsante, Francesco; Mondello, Gabriele; Schlenker, Jean-Marc. A cyclic extension of the earthquake flow I. Geom. Topol. 17 (2013), no. 1, 157--234. doi:10.2140/gt.2013.17.157. https://projecteuclid.org/euclid.gt/1513732522


Export citation

References

  • S I Al'ber, Spaces of mappings into a manifold of negative curvature, Dokl. Akad. Nauk SSSR 178 (1968) 13–16 In Russian; translated in Soviet Math. Dokl. 9 (1968) 6–9
  • L Andersson, T Barbot, R Benedetti, F Bonsante, W M Goldman, F Labourie, K P Scannell, J-M Schlenker, Notes on: “Lorentz spacetimes of constant curvature” [Geom. Dedicata 126 (2007), 3–45; MR2328921] by G. Mess, Geom. Dedicata 126 (2007) 47–70
  • T Barbot, F Béguin, A Zeghib, Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on ${\rm AdS}\sb 3$, Geom. Dedicata 126 (2007) 71–129
  • T Barbot, F Béguin, A Zeghib, Prescribing Gauss curvature of surfaces in $3$–dimensional spacetimes: application to the Minkowski problem in the Minkowski space, Ann. Inst. Fourier (Grenoble) 61 (2011) 511–591
  • M Belraouti, Sur la géométrie de la singularité initiale des espaces-temps plats globalement hyperboliques
  • M Belraouti, personal communication on work in progress (2012)
  • R Benedetti, F Bonsante, Canonical Wick rotations in $3$–dimensional gravity, Mem. Amer. Math. Soc. 198 (2009) viii+164
  • R Benedetti, E Guadagnini, Cosmological time in $(2+1)$–gravity, Nuclear Phys. B 613 (2001) 330–352
  • F Bonsante, J-M Schlenker, Maximal surfaces and the universal Teichmüller space, Invent. Math. 182 (2010) 279–333
  • J Douglas, Minimal surfaces of higher topological structure, Ann. of Math. 40 (1939) 205–298
  • D Dumas, M Wolf, Projective structures, grafting and measured laminations, Geom. Topol. 12 (2008) 351–386
  • J Eells, Jr, J H Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109–160
  • A E Fischer, A J Tromba, A new proof that Teichmüller space is a cell, Trans. Amer. Math. Soc. 303 (1987) 257–262
  • F P Gardiner, W J Harvey, Universal Teichmüller space, from: “Handbook of complex analysis: geometric function theory, Vol. 1”, (R Kühnau, editor), North-Holland, Amsterdam (2002) 457–492
  • P Hartman, On homotopic harmonic maps, Canad. J. Math. 19 (1967) 673–687
  • C D Hodgson, S P Kerckhoff, Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery, J. Differential Geom. 48 (1998) 1–59
  • C D Hodgson, I Rivin, A characterization of compact convex polyhedra in hyperbolic $3$–space, Invent. Math. 111 (1993) 77–111
  • H Jacobowitz, The Gauss-Codazzi equations, Tensor 39 (1982) 15–22
  • J Jost, Two-dimensional geometric variational problems, Pure and Applied Mathematics (New York), John Wiley & Sons Ltd., Chichester (1991)
  • S P Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983) 235–265
  • K Krasnov, J-M Schlenker, Minimal surfaces and particles in $3$–manifolds, Geom. Dedicata 126 (2007) 187–254
  • K Krasnov, J-M Schlenker, On the renormalized volume of hyperbolic $3$–manifolds, Comm. Math. Phys. 279 (2008) 637–668
  • R S Kulkarni, U Pinkall, A canonical metric for Möbius structures and its applications, Math. Z. 216 (1994) 89–129
  • F Labourie, Problème de Minkowski et surfaces à courbure constante dans les variétés hyperboliques, Bull. Soc. Math. France 119 (1991) 307–325
  • F Labourie, Surfaces convexes dans l'espace hyperbolique et ${\bf C}{\rm P}\sp 1$–structures, J. London Math. Soc. 45 (1992) 549–565
  • C Lecuire, J-M Schlenker, The convex core of quasifuchsian manifolds with particles
  • B Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985) 381–386
  • C T McMullen, Complex earthquakes and Teichmüller theory, J. Amer. Math. Soc. 11 (1998) 283–320
  • G Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007) 3–45
  • Y N Minsky, The classification of punctured-torus groups, Ann. of Math. 149 (1999) 559–626
  • G Mondello, Flows of $SL_2(R)$–type on the cotangent space to Teichmüller space, in preparation
  • S Moroianu, J-M Schlenker, Quasi-Fuchsian manifolds with particles, J. Differential Geom. 83 (2009) 75–129
  • J H Sampson, Some properties and applications of harmonic mappings, Ann. Sci. École Norm. Sup. 11 (1978) 211–228
  • K P Scannell, Flat conformal structures and the classification of de Sitter manifolds, Comm. Anal. Geom. 7 (1999) 325–345
  • K P Scannell, M Wolf, The grafting map of Teichmüller space, J. Amer. Math. Soc. 15 (2002) 893–927
  • J-M Schlenker, Métriques sur les polyèdres hyperboliques convexes, J. Differential Geom. 48 (1998) 323–405
  • J-M Schlenker, Hypersurfaces in $H\sp n$ and the space of its horospheres, Geom. Funct. Anal. 12 (2002) 395–435
  • R M Schoen, The role of harmonic mappings in rigidity and deformation problems, from: “Complex geometry”, (G Komatsu, Y Sakane, editors), Lecture Notes in Pure and Appl. Math. 143, Dekker, New York (1993) 179–200
  • R Schoen, S T Yau, On univalent harmonic maps between surfaces, Invent. Math. 44 (1978) 265–278
  • M Spivak, A comprehensive introduction to geometry, Vol. I-V, Publish or Perish (1970)
  • K Tenenblat, On isometric immersions of Riemannian manifolds, Bol. Soc. Brasil. Mat. 2 (1971) 23–36
  • W P Thurston, The geometry and topology of three-manifolds, Princeton Univ. Math. Dept. Lecture Notes (1979) Available at \setbox0\makeatletter\@url http://msri.org/publications/books/gt3m/ {\unhbox0
  • M Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989) 449–479
  • M Wolf, Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space, J. Differential Geom. 33 (1991) 487–539
  • M Wolf, Harmonic maps from surfaces to $\mathbf R$–trees, Math. Z. 218 (1995) 577–593