The Teichmüller space of the Hirsch foliation
[L’espace de Teichmüller du feuilletage de Hirsch]
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 1-51.

Nous prouvons que l’espace de Teichmüller du feuilletage de Hirsch (un feuilletage minimal d’une 3-variété fermée par surfaces hyperboliques non compactes) est homéomorphe à l’espace des courbes fermées du plan. Cela nous permet de prouver que l’espace des métriques hyperboliques sur le feuilletage est un fibré principal trivial. De plus, le groupe structural de ce fibré, i.e. la composante neutre du groupe des homéomorphismes qui sont lisses le long des feuilles et varient transversalement continûment dans la topologie lisse, est contractile.

We prove that the Teichmüller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that the space of hyperbolic metrics on the foliation is a trivial principal fiber bundle. And that the structure group of this bundle, the arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3150
Classification : 57R30, 30F60
Keywords: Teichmüller theory, Riemann surface foliations
Mot clés : Théorie de Teichmülller, feuilletages par surfaces de Riemann
Alvarez, Sébastien 1 ; Lessa, Pablo 2

1 Instituto de Matemática Pura e Aplicada Est. D. Castorina 110 22460-320 Rio de Janeiro (Brazil)
2 IMERL, Facultad de Ingeniería, Universidad de la República Julio Herrera y Reissig 565 CP11300 Montevideo (Uruguay)
@article{AIF_2018__68_1_1_0,
     author = {Alvarez, S\'ebastien and Lessa, Pablo},
     title = {The {Teichm\"uller} space of the {Hirsch} foliation},
     journal = {Annales de l'Institut Fourier},
     pages = {1--51},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3150},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.3150/}
}
TY  - JOUR
AU  - Alvarez, Sébastien
AU  - Lessa, Pablo
TI  - The Teichmüller space of the Hirsch foliation
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 1
EP  - 51
VL  - 68
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.3150/
DO  - 10.5802/aif.3150
LA  - en
ID  - AIF_2018__68_1_1_0
ER  - 
%0 Journal Article
%A Alvarez, Sébastien
%A Lessa, Pablo
%T The Teichmüller space of the Hirsch foliation
%J Annales de l'Institut Fourier
%D 2018
%P 1-51
%V 68
%N 1
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.3150/
%R 10.5802/aif.3150
%G en
%F AIF_2018__68_1_1_0
Alvarez, Sébastien; Lessa, Pablo. The Teichmüller space of the Hirsch foliation. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 1-51. doi : 10.5802/aif.3150. http://www.numdam.org/articles/10.5802/aif.3150/

[1] Ahlfors, Lars V. Lectures on quasiconformal mappings, University Lecture Series, 38, American Mathematical Society, 2006, viii+162 pages (With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard) | MR | Zbl

[2] Ahlfors, Lars V.; Bers, Lipman Riemann’s mapping theorem for variable metrics, Ann. Math., Volume 72 (1960), pp. 385-404 | DOI | MR | Zbl

[3] Alcalde, Fernando; Dal’Bo, Françoise; Martínez, Matilde; Verjovsky, Alberto Minimality of the horocycle flow on foliations by hyperbolic surfaces with non-trivial topology (2014) (https://arxiv.org/abs/1412.3259)

[4] Alessandrini, Daniele; Liu, Lixin; Papadopoulos, Athanase; Su, Weixu On local comparison between various metrics on Teichmüller spaces, Geom. Dedicata, Volume 157 (2012), pp. 91-110 | DOI | MR | Zbl

[5] Alessandrini, Daniele; Liu, Lixin; Papadopoulos, Athanase; Su, Weixu On various Teichmüller spaces of a surface of infinite topological type, Proc. Am. Math. Soc., Volume 140 (2012) no. 2, pp. 561-574 | DOI | MR | Zbl

[6] Alessandrini, Daniele; Liu, Lixin; Papadopoulos, Athanase; Su, Weixu; Sun, Zongliang On Fenchel-Nielsen coordinates on Teichmüller spaces of surfaces of infinite type, Ann. Acad. Sci. Fenn. Math., Volume 36 (2011) no. 2, pp. 621-659 | DOI | MR | Zbl

[7] Angenent, Sigurd B. Nonlinear analytic semiflows, Proc. R. Soc. Edinb., Sect. A, Volume 115 (1990) no. 1-2, pp. 91-107 | DOI | MR | Zbl

[8] Angenent, Sigurd B. Parabolic equations for curves on surfaces. I. Curves with p-integrable curvature, Ann. Math., Volume 132 (1990) no. 3, pp. 451-483 | DOI | MR | Zbl

[9] Baer, Reinhold Kurventypen auf Flächen, J. Reine Angew. Math., Volume 156 (1927), pp. 231-246 | DOI | MR | Zbl

[10] Berger, Melvyn S. Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, J. Differ. Geom., Volume 5 (1971), pp. 325-332 | DOI | MR | Zbl

[11] Bonatti, Christian; Grines, Vyacheslav; Medvedev, Vladislav; Pécou, Elisabeth Topological classification of gradient-like diffeomorphisms on 3-manifolds, Topology, Volume 43 (2004) no. 2, pp. 369-391 | DOI | MR | Zbl

[12] Bonatti, Christian; Paoluzzi, Luisa 3-manifolds which are orbit spaces of diffeomorphisms, Topology, Volume 47 (2008) no. 2, pp. 71-100 | DOI | MR | Zbl

[13] Bonnot, Sylvain Topological model for a class of complex Hénon mappings, Comment. Math. Helv., Volume 81 (2006) no. 4, pp. 827-857 | DOI | MR | Zbl

[14] Buser, Peter Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106, Birkhäuser, 1992, xiv+454 pages | MR | Zbl

[15] Calegari, Danny Foliations and the geometry of 3-manifolds, Oxford Mathematical Monographs, Oxford University Press, 2007, xiv+363 pages | MR | Zbl

[16] Candel, Alberto Uniformization of surface laminations, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993) no. 4, pp. 489-516 | DOI | MR | Zbl

[17] Candel, Alberto; Conlon, Lawrence Foliations. I, Graduate Studies in Mathematics, 23, American Mathematical Society, Providence, RI, 2000, xiv+402 pages | MR | Zbl

[18] Deroin, Bertrand Nonrigidity of hyperbolic surfaces laminations, Proc. Am. Math. Soc., Volume 135 (2007) no. 3, pp. 873-881 | DOI | MR | Zbl

[19] Deroin, Bertrand; Vernicos, Constantin Feuilletage de Hirsch, mesures harmoniques et g-mesures, Publ. Mat. Urug., Volume 12 (2011), pp. 79-85 | MR | Zbl

[20] Dudziak, James J. Vitushkin’s conjecture for removable sets, Universitext, Springer, 2010, xii+331 pages | DOI | MR | Zbl

[21] Earle, Clifford J.; Eells, James jun. A fibre bundle description of Teichmüller theory, J. Differ. Geom., Volume 3 (1969), pp. 19-43 | DOI | MR | Zbl

[22] Earle, Clifford J.; Schatz, Andrea Teichmüller theory for surfaces with boundary, J. Differ. Geom., Volume 4 (1970), pp. 169-185 | DOI | MR | Zbl

[23] Epstein, David Bernard Alper Curves on 2-manifolds and isotopies, Acta Math., Volume 115 (1966), pp. 83-107 | DOI | MR | Zbl

[24] Ghys, Étienne Topologie des feuilles génériques, Ann. Math., Volume 141 (1995) no. 2, pp. 387-422 | DOI | MR | Zbl

[25] Ghys, Étienne Laminations par surfaces de Riemann, Dynamique et géométrie complexes (Lyon, 1997) (Panoramas et Synthèses), Volume 8, Société Mathématique de France, 1999, pp. ix, xi, 49-95 | MR | Zbl

[26] Grayson, Matthew A. Shortening embedded curves, Ann. Math., Volume 129 (1989) no. 1, pp. 71-111 | DOI | MR | Zbl

[27] Hatcher, A. Notes on basic 3-manifolds topology (2007) (xii+195 pages, available online at http://www.math.cornell.edu/~hatcher/3M/3Mfds.pdf)

[28] Hirsch, Morris W. A stable analytic foliation with only exceptional minimal sets, Dynamical systems (Warwick 1974) (Lecture Notes in Math.), Volume 468, Springer, 1975, pp. 9-10 | MR | Zbl

[29] Hubbard, John H.; Oberste-Vorth, Ralph W. Hénon mappings in the complex domain. I. The global topology of dynamical space, Publ. Math., Inst. Hautes Étud. Sci., Volume 79 (1994), pp. 5-46 | DOI | MR | Zbl

[30] Iglesias, Jorge; Portela, Aldo; Rovella, Alvaro C 1 stable maps: examples without saddles, Fundam. Math., Volume 208 (2010) no. 1, pp. 23-33 | DOI | MR | Zbl

[31] Imayoshi, Yoichi; Taniguchi, Masahiko An introduction to Teichmüller spaces, Springer, 1992, xiv+279 pages (Translated and revised from the Japanese by the authors) | DOI | MR | Zbl

[32] Kahn, Jeremy; Markovic, Vladimir The good pants homology and the Ehrenpreis Conjecture, Ann. Math., Volume 182 (2015) no. 1, pp. 1-72 | DOI | MR | Zbl

[33] Mantegazza, Carlo Lecture notes on mean curvature flow, Progress in Mathematics, 290, Birkhäuser, 2011, xii+166 pages | DOI | MR | Zbl

[34] Moore, Calvin C.; Schochet, Claude Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications, 9, Cambridge University Press, 2006, xiv+293 pages | MR | Zbl

[35] Penner, Robert C.; Šarić, Dragomir Teichmüller theory of the punctured solenoid, Geom. Dedicata, Volume 132 (2008), pp. 179-212 | DOI | MR | Zbl

[36] Šarić, Dragomir The Teichmüller theory of the solenoid, Handbook of Teichmüller theory. Vol. II (IRMA Lectures in Mathematics and Theoretical Physics), Volume 13, European Mathematical Society (EMS), 2009, pp. 811-857 | DOI | MR | Zbl

[37] Schwartz, Richard A projectively natural flow for circle diffeomorphisms, Invent. Math., Volume 110 (1992) no. 3, pp. 627-647 | DOI | MR | Zbl

[38] Sullivan, Dennis Bounds, quadratic differentials, and renormalization conjectures, Mathematics into the twenty-first century. (Providence, RI, 1988) (American Mathematical Society Centennial Publications), Volume 2, American Mathematical Society, 1992, pp. 417-466 | MR | Zbl

[39] Sullivan, Dennis Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Inc., 1993, pp. 543-564 | MR | Zbl

[40] Sullivan, Dennis Solenoidal manifolds, J. Singul., Volume 9 (2014), pp. 203-205 | MR | Zbl

[41] Verjovsky, Alberto Commentaries on the paper Solenoidal manifolds by Dennis Sullivan, J. Singul., Volume 9 (2014), pp. 245-251 | MR | Zbl

Cité par Sources :