Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



AdS 3-manifolds and Higgs bundles

Authors: Daniele Alessandrini and Qiongling Li
Journal: Proc. Amer. Math. Soc. 146 (2018), 845-860
MSC (2010): Primary 57M20, 53C07; Secondary 58E12, 58E20
Published electronically: October 23, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we investigate the relationships between closed AdS $ 3$-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for their volume.

We give natural foliations of the AdS structure with time-like geodesic circles and we use these circles to construct equivariant minimal immersions of the Poincaré disc into the Grassmannian of time-like 2-planes of $ \mathbb{R}^{2,2}$.

References [Enhancements On Off] (What's this?)

  • [1] D. Baraglia, $ G_2$ geometry and integrable systems, Ph.D. Thesis, 2009, arXiv:1002.1767.
  • [2] T. Barbot, F. Bonsante, J. Danciger, W. Goldman, F. Guéritaud, F. Kassel, K. Krasnov, J.-M. Schlenker, and A. Zeghib, Some open questions on Anti-de Sitter geometry, arXiv:1205.6103.
  • [3] Marc Burger, Alessandra Iozzi, and Anna Wienhard, Higher Teichmüller spaces: from $ {\rm SL}(2,\mathbb{R})$ to other Lie groups, Handbook of Teichmüller theory. Vol. IV, IRMA Lect. Math. Theor. Phys., vol. 19, Eur. Math. Soc., Zürich, 2014, pp. 539-618. MR 3289711,
  • [4] Kevin Corlette, Flat $ G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), no. 3, 361-382. MR 965220
  • [5] Bertrand Deroin and Nicolas Tholozan, Dominating surface group representations by Fuchsian ones, Int. Math. Res. Not. IMRN 13 (2016), 4145-4166. MR 3544632,
  • [6] S. K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. (3) 55 (1987), no. 1, 127-131. MR 887285,
  • [7] W. Goldman, Geometric structures and varieties of representations, preprint.
  • [8] William M. Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988), no. 3, 557-607. MR 952283,
  • [9] P. Gothen, The topology of the Higgs bundle moduli space, Ph.D. Thesis, 1995.
  • [10] F. Gueritaud and F. Kassel, Maximally stretched laminations on geometrically finite hyperbolic manifolds, to appear in Geom. Topol.
  • [11] Frédéric Hélein and John C. Wood, Harmonic maps, Handbook of global analysis, Elsevier Sci. B. V., Amsterdam, 2008, pp. 417-491, 1213. MR 2389639,
  • [12] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59-126. MR 887284,
  • [13] N. J. Hitchin, Lie groups and Teichmüller space, Topology 31 (1992), no. 3, 449-473. MR 1174252,
  • [14] F. Kassel, Quotients compacts d'espaces homogènes réels ou p-adiques, Ph.D. Thesis, 2009.
  • [15] Bruno Klingler, Complétude des variétés lorentziennes à courbure constante, Math. Ann. 306 (1996), no. 2, 353-370 (French). MR 1411352,
  • [16] Ravi S. Kulkarni and Frank Raymond, $ 3$-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom. 21 (1985), no. 2, 231-268. MR 816671
  • [17] F. Labourie, Chern-Simons invariant and Tholozan volume formula, Seminar talk at MSRI,
  • [18] François Salein, Variétés anti-de Sitter de dimension 3 exotiques, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 1, 257-284 (French, with English and French summaries). MR 1762345
  • [19] Carlos T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867-918. MR 944577,
  • [20] Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5-95. MR 1179076
  • [21] N. Tholozan, Dominating surface group representations and deforming closed Anti-de Sitter 3-manifolds, to appear in Geom. Topol.
  • [22] N. Tholozan, Uniformisation des variétés pseudo-riemanniennes localement homogènes, Ph.D. Thesis, 2014.
  • [23] N. Tholozan, The volumes of complete Anti-de Sitter $ 3$-manifolds, arXiv:1509.04178.
  • [24] William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy. MR 1435975
  • [25] Francisco Torralbo, Minimal Lagrangian immersions in $ \mathbb{RH}^2\times\mathbb{RH}^2$, Symposium on the Differential Geometry of Submanifolds, [s.n.], [s.l.], 2007, pp. 217-219. MR 2509502
  • [26] Michael Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449-479. MR 982185

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57M20, 53C07, 58E12, 58E20

Retrieve articles in all journals with MSC (2010): 57M20, 53C07, 58E12, 58E20

Additional Information

Daniele Alessandrini
Affiliation: Mathematisches Institut, Universitaet Heidelberg, INF 205, 69120, Heidelberg, Germany

Qiongling Li
Affiliation: Centre for Quantum Geometry of Moduli Spaces (QGM), Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark – and – Department of Mathematics, California Institute of Technology, 1200 East California Boulevard, Pasadena, California 91125

Received by editor(s): October 28, 2015
Received by editor(s) in revised form: November 9, 2015, September 25, 2016, and November 14, 2016
Published electronically: October 23, 2017
Additional Notes: This work was started when both authors were visiting MSRI. Research at MSRI was supported in part by NSF grant DMS-0441170. Both authors acknowledge the support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). The second author was supported by the center of excellence grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95).
Communicated by: Michael Wolf
Article copyright: © Copyright 2017 Daniele Alessandrini and Qiongling Li

American Mathematical Society