## AdS 3-manifolds and Higgs bundles

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- by Daniele Alessandrini and Qiongling Li PDF
- Proc. Amer. Math. Soc.
**146**(2018), 845-860

## 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

- D. Baraglia,
*$G_2$ geometry and integrable systems*, Ph.D. Thesis, 2009, arXiv:1002.1767. - 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. - Marc Burger, Alessandra Iozzi, and Anna Wienhard,
*Higher Teichmüller spaces: from $\textrm {SL}(2,\Bbb R)$ 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**, DOI 10.4171/117-1/14 - Kevin Corlette,
*Flat $G$-bundles with canonical metrics*, J. Differential Geom.**28**(1988), no. 3, 361–382. MR**965220** - Bertrand Deroin and Nicolas Tholozan,
*Dominating surface group representations by Fuchsian ones*, Int. Math. Res. Not. IMRN**13**(2016), 4145–4166. MR**3544632**, DOI 10.1093/imrn/rnv275 - S. K. Donaldson,
*Twisted harmonic maps and the self-duality equations*, Proc. London Math. Soc. (3)**55**(1987), no. 1, 127–131. MR**887285**, DOI 10.1112/plms/s3-55.1.127 - W. Goldman,
*Geometric structures and varieties of representations*, preprint. - William M. Goldman,
*Topological components of spaces of representations*, Invent. Math.**93**(1988), no. 3, 557–607. MR**952283**, DOI 10.1007/BF01410200 - P. Gothen,
*The topology of the Higgs bundle moduli space*, Ph.D. Thesis, 1995. - F. Gueritaud and F. Kassel,
*Maximally stretched laminations on geometrically finite hyperbolic manifolds*, to appear in Geom. Topol. - Frédéric Hélein and John C. Wood,
*Harmonic maps*, Handbook of global analysis, Elsevier Sci. B. V., Amsterdam, 2008, pp. 417–491, 1213. MR**2389639**, DOI 10.1016/B978-044452833-9.50009-7 - N. J. Hitchin,
*The self-duality equations on a Riemann surface*, Proc. London Math. Soc. (3)**55**(1987), no. 1, 59–126. MR**887284**, DOI 10.1112/plms/s3-55.1.59 - N. J. Hitchin,
*Lie groups and Teichmüller space*, Topology**31**(1992), no. 3, 449–473. MR**1174252**, DOI 10.1016/0040-9383(92)90044-I - F. Kassel,
*Quotients compacts d’espaces homogènes réels ou p-adiques*, Ph.D. Thesis, 2009. - Bruno Klingler,
*Complétude des variétés lorentziennes à courbure constante*, Math. Ann.**306**(1996), no. 2, 353–370 (French). MR**1411352**, DOI 10.1007/BF01445255 - 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** - F. Labourie,
*Chern-Simons invariant and Tholozan volume formula*, Seminar talk at MSRI, https://www.msri.org/seminars/21432. - 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**, DOI 10.5802/aif.1754 - 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**, DOI 10.1090/S0894-0347-1988-0944577-9 - Carlos T. Simpson,
*Higgs bundles and local systems*, Inst. Hautes Études Sci. Publ. Math.**75**(1992), 5–95. MR**1179076**, DOI 10.1007/BF02699491 - N. Tholozan,
*Dominating surface group representations and deforming closed Anti-de Sitter 3-manifolds*, to appear in Geom. Topol. - N. Tholozan,
*Uniformisation des variétés pseudo-riemanniennes localement homogènes*, Ph.D. Thesis, 2014. - N. Tholozan,
*The volumes of complete Anti-de Sitter $3$-manifolds*, arXiv:1509.04178. - 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**, DOI 10.1515/9781400865321 - Francisco Torralbo,
*Minimal Lagrangian immersions in $\Bbb {RH}^2\times \Bbb {RH}^2$*, Symposium on the Differential Geometry of Submanifolds, [s.n.], [s.l.], 2007, pp. 217–219. MR**2509502** - Michael Wolf,
*The Teichmüller theory of harmonic maps*, J. Differential Geom.**29**(1989), no. 2, 449–479. MR**982185**

## Additional Information

**Daniele Alessandrini**- Affiliation: Mathematisches Institut, Universitaet Heidelberg, INF 205, 69120, Heidelberg, Germany
- MR Author ID: 849772
- Email: daniele.alessandrini@gmail.com
**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
- MR Author ID: 1149454
- Email: qiongling.li@gmail.com
- 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
- © Copyright 2017 Daniele Alessandrini and Qiongling Li
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 845-860 - MSC (2010): Primary 57M20, 53C07; Secondary 58E12, 58E20
- DOI: https://doi.org/10.1090/proc/13586
- MathSciNet review: 3731716