These notes are an extended version of a talk given by the author in the seminar Théorie Spectrale et Géométrie at the Institut Fourier in November 2016. We present here some aspects of a work in collaboration with B. Collier and N. Tholozan [9]. We describe how Higgs bundle theory and pseudo-hyperbolic geometry interfere in the study of maximal representations into Hermitian Lie groups of rank .
@article{TSG_2016-2017__34__97_0, author = {J\'er\'emy Toulisse}, title = {Higgs bundles, pseudo-hyperbolic geometry and maximal representations}, journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie}, pages = {97--114}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {34}, year = {2016-2017}, doi = {10.5802/tsg.357}, zbl = {1359.53053}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.357/} }
TY - JOUR AU - Jérémy Toulisse TI - Higgs bundles, pseudo-hyperbolic geometry and maximal representations JO - Séminaire de théorie spectrale et géométrie PY - 2016-2017 SP - 97 EP - 114 VL - 34 PB - Institut Fourier PP - Grenoble UR - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.357/ DO - 10.5802/tsg.357 LA - en ID - TSG_2016-2017__34__97_0 ER -
%0 Journal Article %A Jérémy Toulisse %T Higgs bundles, pseudo-hyperbolic geometry and maximal representations %J Séminaire de théorie spectrale et géométrie %D 2016-2017 %P 97-114 %V 34 %I Institut Fourier %C Grenoble %U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.357/ %R 10.5802/tsg.357 %G en %F TSG_2016-2017__34__97_0
Jérémy Toulisse. Higgs bundles, pseudo-hyperbolic geometry and maximal representations. Séminaire de théorie spectrale et géométrie, Volume 34 (2016-2017), pp. 97-114. doi : 10.5802/tsg.357. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.357/
[1] Daniele Alessandrini; Brian Collier The geometry of maximal components of the character variety (2017) (https://arxiv.org/abs/1708.05361, to appear in Geom. Topol.) | Zbl
[2] Thierry Barbot; François Béguin; Abdelghani Zeghib Feuilletages des espaces temps globalement hyperboliques par des hypersurfaces à courbure moyenne constante, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 3, pp. 245-250 | DOI | MR | Zbl
[3] Francesco Bonsante; Jean-Marc Schlenker Maximal surfaces and the universal Teichmüller space, Invent. Math., Volume 182 (2010) no. 2, pp. 279-333 | Zbl
[4] Steven B. Bradlow; Oscar García-Prada; Peter B. Gothen Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces, Geom. Dedicata, Volume 122 (2006), pp. 185-213 | DOI | MR | Zbl
[5] Marc Burger; Alessandra Iozzi; François Labourie; Anna Wienhard Maximal representations of surface groups: symplectic Anosov structures, Pure Appl. Math. Q., Volume 1 (2005) no. 3, pp. 543-590 | DOI | MR | Zbl
[6] Marc Burger; Alessandra Iozzi; Anna Wienhard Surface group representations with maximal Toledo invariant, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 5, pp. 387-390 | DOI | MR | Zbl
[7] Marc Burger; Alessandra Iozzi; Anna Wienhard Surface group representations with maximal Toledo invariant, Ann. Math., Volume 172 (2010) no. 1, pp. 517-566 | DOI | MR | Zbl
[8] Brian Collier Finite order automorphisms of Higgs bundles: theory and application, University of Illinois Urbana Champaign (USA) (2016) (Ph. D. Thesis) | Zbl
[9] Brian Collier; Nicolas Tholozan; Jérémy Toulisse The geometry of maximal representations of surface groups into (2017) (https://arxiv.org/abs/1702.08799, to appear in Duke Math. J.)
[10] Kevin Corlette Flat -bundles with canonical metrics, J. Differ. Geom., Volume 28 (1988) no. 3, pp. 361-382 | MR | Zbl
[11] James Eells; Joseph H. Sampson Harmonic mappings of Riemannian manifolds, Am. J. Math., Volume 86 (1964), pp. 109-160 | DOI | MR | Zbl
[12] Oscar García-Prada; Ignasi Mundet i Riera Representations of the fundamental group of a closed oriented surface in , Topology, Volume 43 (2004) no. 4, pp. 831-855 | DOI | MR | Zbl
[13] Peter B. Gothen Components of spaces of representations and stable triples, Topology, Volume 40 (2001) no. 4, pp. 823-850 | DOI | MR | Zbl
[14] Olivier Guichard; Anna Wienhard Anosov representations: domains of discontinuity and applications, Invent. Math., Volume 190 (2012) no. 2, pp. 357-438 | DOI | MR | Zbl
[15] Nigel J. Hitchin The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., Volume 55 (1987) no. 1, pp. 59-126 | DOI | MR | Zbl
[16] T. Ishihara The harmonic Gauss maps in a generalized sense, J. Lond. Math. Soc., Volume 26 (1982) no. 1, pp. 104-112 | DOI | MR | Zbl
[17] Kirill Krasnov; Jean-Marc Schlenker Minimal surfaces and particles in 3-manifolds, Geom. Dedicata, Volume 126 (2007), pp. 187-254 | DOI | MR | Zbl
[18] François Labourie Anosov flows, surface groups and curves in projective space, Invent. Math., Volume 165 (2006) no. 1, pp. 51-114 | DOI | MR | Zbl
[19] François Labourie Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q., Volume 3 (2007) no. 4, pp. 1057-1099 | DOI | MR | Zbl
[20] François Labourie Cross ratios, Anosov representations and the energy functional on Teichmüller space, Ann. Sci. Éc. Norm. Supér., Volume 41 (2008) no. 3, pp. 437-469 | MR | Zbl
[21] François Labourie Cyclic surfaces and Hitchin components in rank 2, Ann. Math., Volume 185 (2017) no. 1, pp. 1-58 | MR | Zbl
[22] John C. Loftin Affine spheres and convex -manifolds, Am. J. Math., Volume 123 (2001) no. 2, pp. 255-274 | MR | Zbl
[23] Geoffrey Mess Lorentz spacetimes of constant curvature, Geom. Dedicata, Volume 126 (2007), pp. 3-45 | DOI | MR | Zbl
[24] John W. Milnor On the existence of a connection with curvature zero, Comment. Math. Helv., Volume 32 (1958), pp. 215-223 | DOI | MR | Zbl
[25] Richard M. Schoen The role of harmonic mappings in rigidity and deformation problems, Complex geometry (Osaka, 1990) (Lecture Notes in Pure and Applied Mathematics), Volume 143, Marcel Dekker, 1993, pp. 179-200 | MR | Zbl
[26] Carlos T. Simpson Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Am. Math. Soc., Volume 1 (1988) no. 4, pp. 867-918 | DOI | MR | Zbl
[27] Carlos T. Simpson Moduli of representations of the fundamental group of a smooth projective variety. I, Publ. Math., Inst. Hautes Étud. Sci. (1994) no. 79, pp. 47-129 | DOI | MR | Zbl
[28] Carlos T. Simpson Moduli of representations of the fundamental group of a smooth projective variety. II, Publ. Math., Inst. Hautes Étud. Sci. (1995) no. 80, pp. 5-79 | MR | Zbl
[29] Carlos T. Simpson Katz’s middle convolution algorithm, Pure Appl. Math. Q., Volume 5 (2009) no. 2, pp. 781-852 | DOI | MR | Zbl
Cited by Sources: