Cross ratios, Anosov representations and the energy functional on Teichmüller space
[Birapports, représentations Anosov et la fonctionnelle d'énergie sur les espaces de Teichmüller]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 439-471.

Nous étudions deux classes de représentations linéaires d'un groupe de surface  : les représentations de Hitchin et les représentations symplectiques maximales. En reliant ces représentations à des birapports, nous montrons qu'elles sont déplaçantes, c'est-à-dire que leurs longueurs de translation sont grossièrement contrôlées par celles du graphe de Cayley. Ceci nous permet de montrer que le groupe modulaire agit proprement sur l'espace de ces représentations et que la fonctionnelle énergie associée à une telle représentation est propre. Nous en déduisons alors l'existence de surfaces minimales dans les quotients d'espaces symétriques associés et en tirons deux conséquences  : un résultat de rigidité pour les représentations symplectiques et un résultat partiel concernant la description de la composante de Hitchin en termes purement holomorphes.

We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence of minimal surfaces in the quotient of the associated symmetric spaces, a fact which leads to two consequences: a rigidity result for maximal symplectic representations and a partial result concerning a purely holomorphic description of the Hichin component.

DOI : 10.24033/asens.2072
Classification : 32G15, 53C43, 20H10, 49Q05
Keywords: Hitchin components, energy, cross ratio, Toledo invariant, harmonic mappings, minimal surfaces
Mot clés : composantes de Hitchin, énergie, cross ratio, invariant de Toledo, harmonic mappings, surfaces minimales
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     author = {Labourie, Fran\c{c}ois},
     title = {Cross ratios, {Anosov} representations and the energy functional on {Teichm\"uller} space},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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Labourie, François. Cross ratios, Anosov representations and the energy functional on Teichmüller space. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 439-471. doi : 10.24033/asens.2072. http://archive.numdam.org/articles/10.24033/asens.2072/

[1] F. Bonahon, The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988), 139-162. | MR | Zbl

[2] R. Bowen, The equidistribution of closed geodesics, Amer. J. Math. 94 (1972), 413-423. | MR | Zbl

[3] S. B. Bradlow, O. García-Prada & P. B. Gothen, Surface group representations and U(p,q)-Higgs bundles, J. Differential Geom. 64 (2003), 111-170. | Zbl

[4] S. B. Bradlow, O. García-Prada & P. B. Gothen, Moduli spaces of holomorphic triples over compact Riemann surfaces, Math. Ann. 328 (2004), 299-351. | Zbl

[5] S. B. Bradlow, O. García-Prada & P. B. Gothen, Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces, Geom. Dedicata 122 (2006), 185-213. | Zbl

[6] M. Burger & A. Iozzi, Bounded Kähler class rigidity of actions on Hermitian symmetric spaces, Ann. Sci. École Norm. Sup. 37 (2004), 77-103. | Numdam | Zbl

[7] M. Burger, A. Iozzi, F. Labourie & A. Wienhard, Maximal representations of surface groups: symplectic Anosov structures, Pure Appl. Math. Q. 1 (2005), 543-590. | Zbl

[8] M. Burger, A. Iozzi & A. Wienhard, Surface group representations with maximal Toledo invariant, C. R. Math. Acad. Sci. Paris 336 (2003), 387-390. | Zbl

[9] M. Burger, A. Iozzi & A. Wienhard, Hermitian symmetric spaces and Kähler rigidity, preprint, 2006. | Zbl

[10] S. Choi & W. M. Goldman, Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993), 657-661. | Zbl

[11] K. Corlette, Flat G-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361-382. | MR | Zbl

[12] C. Croke & A. Fathi, An inequality between energy and intersection, Bull. London Math. Soc. 22 (1990), 489-494. | Zbl

[13] T. Delzant, O. Guichard, F. Labourie & S. Mozes, Well displacing representations and orbit maps, preprint, 2007.

[14] A. Domic & D. Toledo, The Gromov norm of the Kähler class of symmetric domains, Math. Ann. 276 (1987), 425-432. | Zbl

[15] S. K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987), 127-131. | MR | Zbl

[16] V. Fock & A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1-211. | Numdam | Zbl

[17] O 43 (2004), 831-855. | Zbl

[18] W. M. Goldman, Discontinuous group and the Euler class, Thèse, University of Berkeley, California, 1980. | MR

[19] W. M. Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988), 557-607. | MR | Zbl

[20] W. M. Goldman, Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), 791-845. | MR | Zbl

[21] W. M. Goldman & R. A. Wentworth, Energy of twisted harmonic maps of Riemann surfaces, in In the tradition of Ahlfors-Bers. IV, Contemp. Math. 432, Amer. Math. Soc., 2007, 45-61. | Zbl

[22] P. B. Gothen, Components of spaces of representations and stable triples, Topology 40 (2001), 823-850. | MR | Zbl

[23] O. Guichard, Composantes de Hitchin et représentations hyperconvexes de groupes de surface, preprint, to appear in J. Diff. Geom., 2005. | Zbl

[24] R. D. I. Gulliver, R. Osserman & H. L. Royden, A theory of branched immersions of surfaces, Amer. J. Math. 95 (1973), 750-812. | Zbl

[25] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126. | MR | Zbl

[26] N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), 449-473. | MR | Zbl

[27] F. Labourie, Existence d'applications harmoniques tordues à valeurs dans les variétés à courbure négative, Proc. Amer. Math. Soc. 111 (1991), 877-882. | MR | Zbl

[28] F. Labourie, ℝℙ 2 -structures et différentielles cubiques holomorphes, in GARC Conference in Differential Geometry, Seoul National University, 1997.

[29] F. Labourie, Cross ratios, surface groups, SL(n,) and diffeomorphisms of the circle, preprint arXiv:math.DG/0512070, to appear in Inst. Hautes Études Sci. Publ. Math., 2005. | MR | Zbl

[30] F. Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006), 51-114. | MR | Zbl

[31] F. Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, preprint arXiv:math.DG/0611250, to appear in Pure Appl. Math. Q., 2006. | MR | Zbl

[32] F. Labourie & G. Mcshane, Cross ratios and identities for higher Teichmüller-Thurston theory, preprint arXiv:math.DG/0611245, 2006. | Zbl

[33] F. Ledrappier, Structure au bord des variétés à courbure négative, in Séminaire de Théorie Spectrale et Géométrie, No. 13, Année 1994-1995, Sémin. Théor. Spectr. Géom. 13, Univ. Grenoble I, 1995, 97-122. | Numdam | MR | Zbl

[34] J. C. Loftin, Affine spheres and convex ℝℙ n -manifolds, Amer. J. Math. 123 (2001), 255-274. | MR | Zbl

[35] C. T. Mcmullen, Teichmüller theory notes, http://www.math.harvard.edu/~ctm/home/text/class/harvard/275/05/html/home/course/course.pdf, 2006.

[36] J. Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv. 32 (1958), 215-223. | MR | Zbl

[37] J.-P. Otal, Sur la géometrie symplectique de l'espace des géodésiques d'une variété à courbure négative, Rev. Mat. Iberoamericana 8 (1992), 441-456. | MR | Zbl

[38] J. Sacks & K. Uhlenbeck, The existence of minimal immersions of 2-spheres, Ann. of Math. 113 (1981), 1-24. | Zbl

[39] J. Sacks & K. Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), 639-652. | Zbl

[40] R. Schoen & S. T. Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. 110 (1979), 127-142. | Zbl

[41] D. Toledo, Harmonic maps from surfaces to certain Kähler manifolds, Math. Scand. 45 (1979), 13-26. | MR | Zbl

[42] D. Toledo, Representations of surface groups in complex hyperbolic space, J. Differential Geom. 29 (1989), 125-133. | MR | Zbl

[43] V. G. Turaev, A cocycle of the symplectic first Chern class and Maslov indices, Funktsional. Anal. i Prilozhen. 18 (1984), 43-48. | MR | Zbl

[44] A. Wienhard, Bounded cohomology and geometry, Thèse, Universität Bonn, 2004. | MR | Zbl

[45] E. Z. Xia, The moduli of flat U(p,1) structures on Riemann surfaces, Geom. Dedicata 97 (2003), 33-43, Special volume dedicated to the memory of Hanna Miriam Sandler (1960-1999). | MR | Zbl

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