Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 039, 32 pages      arXiv:1809.07290      https://doi.org/10.3842/SIGMA.2019.039
Contribution to the Special Issue on Geometry and Physics of Hitchin Systems

Higgs Bundles and Geometric Structures on Manifolds

Daniele Alessandrini
Ruprecht-Karls-Universitaet Heidelberg, INF 205, 69120, Heidelberg, Germany

Received September 28, 2018, in final form April 17, 2019; Published online May 10, 2019

Abstract
Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichmüller theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Ph.D. Thesis that Higgs bundles can also be used to construct geometric structures in some interesting cases. In this paper, we will explain the main ideas behind this theory and we will survey some recent results in this direction, which are joint work with Qiongling Li.

Key words: geometric structures; Higgs bundles; higher Teichmüller theory; Anosov representations.

pdf (532 kb)   tex (38 kb)  

References

  1. Alessandrini D., Collier B., The geometry of maximal components of the ${\rm PSp}(4,{\mathbb R})$ character variety, Geom. Topol., to appear, arXiv:1708.05361.
  2. Alessandrini D., Li Q., AdS 3-manifolds and Higgs bundles, Proc. Amer. Math. Soc. 146 (2018), 845-860, arXiv:1510.07745.
  3. Alessandrini D., Li Q., The nilpotent cone for ${\rm PSL}(2,{\mathbb C})$-Higgs bundles, in preparation.
  4. Alessandrini D., Li Q., Projections from flags manifolds to the hyperbolic plane, in preparation.
  5. Alessandrini D., Li Q., Projective structures with (quasi-)Hitchin holonomy, in preparation.
  6. Alessandrini D., Maloni S., Wienhard A., The geometry of quasi-Hitchin symplectic representations, in preparation.
  7. Baba S., $2\pi$-grafting and complex projective structures, I, Geom. Topol. 19 (2015), 3233-3287, arXiv:1011.5051.
  8. Baba S., $2\pi$-grafting and complex projective structures with generic holonomy, Geom. Funct. Anal. 27 (2017), 1017-1069, arXiv:1307.2310.
  9. Baraglia D., ${G}_2$ geometry and integrable systems, Ph.D. Thesis, University of Oxford, 2009, arXiv:1002.1767.
  10. Baues O., The deformation of flat affine structures on the two-torus, in Handbook of Teichmüller Theory, Vol. IV, IRMA Lect. Math. Theor. Phys., Vol. 19, Eur. Math. Soc., Zürich, 2014, 461-537, arXiv:1112.3263.
  11. Choi S., Goldman W.M., Convex real projective structures on closed surfaces are closed, Proc. Amer. Math. Soc. 118 (1993), 657-661.
  12. Choi S., Goldman W.M., The classification of real projective structures on compact surfaces, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 161-171.
  13. Collier B., Li Q., Asymptotics of Higgs bundles in the Hitchin component, Adv. Math. 307 (2017), 488-558, arXiv:1405.1106.
  14. Collier B., Tholozan N., Toulisse J., The geometry of maximal representations of surface groups into ${\rm SO}(2,n)$, Duke Math. J., to appear, arXiv:1702.08799.
  15. Dumas D., Sanders A., Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations, arXiv:1704.01091.
  16. Fock V., Goncharov A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes 'Etudes Sci. (2006), 1-211, arXiv:math.AG/0311149.
  17. Gallo D., Kapovich M., Marden A., The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. of Math. 151 (2000), 625-704.
  18. Goldman W.M., Discontinuous groups and the Euler class, Ph.D. Thesis, University of California, Berkeley, 1980.
  19. Goldman W.M., Geometric structures and varieties of representations, available at https://www.math.stonybrook.edu/~mlyubich/Archive/Geometry/Hyperbolic%20Geometry/Goldman.pdf.
  20. Goldman W.M., Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), 297-326.
  21. Goldman W.M., Convex real projective structures on compact surfaces, J. Differential Geom. 31 (1990), 791-845.
  22. Gromov M., Hyperbolic groups, in Essays in Group Theory, Math. Sci. Res. Inst. Publ., Vol. 8, Springer, New York, 1987, 75-263.
  23. Guichard O., Wienhard A., Convex foliated projective structures and the Hitchin component for ${\rm PSL}_4({\bf R})$, Duke Math. J. 144 (2008), 381-445, arXiv:math.DG/0702184.
  24. Guichard O., Wienhard A., Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012), 357-438, arXiv:1108.0733.
  25. Helgason S., Differential geometry, Lie groups, and symmetric spaces,Graduate Studies in Mathematics, Vol. 34, Amer. Math. Soc., Providence, RI, 2001.
  26. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  27. Hitchin N.J., Lie groups and Teichmüller space, Topology 31 (1992), 449-473.
  28. Humphreys J.E., Linear algebraic groups, Graduate Texts in Mathematics, Vol. 21, Springer-Verlag, New York - Heidelberg, 1975.
  29. Kapovich M., Leeb B., Porti J., Dynamics on flag manifolds: domains of proper discontinuity and cocompactness, Geom. Topol. 22 (2018), 157-234, arXiv:1306.3837.
  30. Labourie F., Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006), 51-114, arXiv:math.DG/0401230.
  31. Labourie F., Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007), 1057-1099, arXiv:math.DG/0611250.
  32. Labourie F., Cross ratios, Anosov representations and the energy functional on Teichmüller space, Ann. Sci. 'Ec. Norm. Supér. (4) 41 (2008), 437-469, arXiv:math.DG/0512070.
  33. Loftin J.C., Affine spheres and convex $\mathbb{RP}^n$-manifolds, Amer. J. Math. 123 (2001), 255-274.
  34. Sampson J.H., Some properties and applications of harmonic mappings, Ann. Sci. 'Ecole Norm. Sup. (4) 11 (1978), 211-228.
  35. Sanders A.M., Minimal surfaces, hyperbolic 3-manifolds, and related deformation spaces, Ph.D. Thesis, University of Maryland, College Park, 2013.
  36. Serre J.-P., Lie algebras and Lie groups, Lecture Notes in Math., Vol. 1500, Springer-Verlag, Berlin, 2006.
  37. Tholozan N., Dominating surface group representations and deforming closed anti-de Sitter 3-manifolds, Geom. Topol. 21 (2017), 193-214, arXiv:1403.7479.
  38. Tholozan N., The volume of complete anti-de Sitter 3-manifolds, J. Lie Theory 28 (2018), 619-642, arXiv:1509.04178.
  39. Thurston W.P., Three-dimensional geometry and topology, Vol. 1, Princeton Mathematical Series, Vol. 35, Princeton University Press, Princeton, NJ, 1997.
  40. Toledo D., Representations of surface groups in complex hyperbolic space, J. Differential Geom. 29 (1989), 125-133.
  41. Uhlenbeck K.K., Closed minimal surfaces in hyperbolic $3$-manifolds, in Seminar on Minimal Submanifolds, Ann. of Math. Stud., Vol. 103, Princeton University Press, Princeton, NJ, 1983, 147-168.
  42. Wolf M., The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), 449-479.

Previous article  Next article  Contents of Volume 15 (2019)