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

SIGMA 15 (2019), 035, 30 pages      arXiv:1809.05747
Contribution to the Special Issue on Geometry and Physics of Hitchin Systems

An Introduction to Higgs Bundles via Harmonic Maps

Qiongling Li
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Received October 16, 2018, in final form April 26, 2019; Published online May 04, 2019

This survey studies equivariant harmonic maps arising from Higgs bundles. We explain the non-abelian Hodge correspondence and focus on the role of equivariant harmonic maps in the correspondence. With the preparation, we review current progress towards some open problems in the study of equivariant harmonic maps.

Key words: Higgs bundles; harmonic maps; non-abelian Hodge correspondence.

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  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. Baraglia D., ${G}_2$ geometry and integrable systems, Ph.D. Thesis, University of Oxford, 2009, arXiv:1002.1767.
  3. Baraglia D., Cyclic Higgs bundles and the affine Toda equations, Geom. Dedicata 174 (2015), 25-42, arXiv:1011.6421.
  4. Bestvina M., Degenerations of the hyperbolic space, Duke Math. J. 56 (1988), 143-161.
  5. Bradlow S.B., García-Prada O., Gothen P.B., Deformations of maximal representations in ${\rm Sp}(4,\mathbb{R})$, Q. J. Math. 63 (2012), 795-843, arXiv:0903.5496.
  6. Burger M., Iozzi A., Labourie F., Wienhard A., Maximal representations of surface groups: symplectic Anosov structures, Pure Appl. Math. Q. 1 (2005), 543-590, arXiv:math.DG/0506079.
  7. Burger M., Iozzi A., Wienhard A., Surface group representations with maximal Toledo invariant, Ann. of Math. 172 (2010), 517-566, arXiv:math.DG/0605656.
  8. Collier B., Maximal ${\rm Sp}(4,\mathbb{R})$ surface group representations, minimal immersions and cyclic surfaces, Geom. Dedicata 180 (2016), 241-285, arXiv:1503.03526.
  9. Collier B., Li Q., Asymptotics of Higgs bundles in the Hitchin component, Adv. Math. 307 (2017), 488-558, arXiv:1405.1106.
  10. Collier B., Nicolas T., Toulisse J., The geometry of maximal representations of surface groups into ${\rm SO}(2,n)$, Duke Math. J., to appear, arXiv:1702.08799.
  11. Corlette K., Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361-382.
  12. Dai S., Li Q., Minimal surfaces for Hitchin representations, J. Differential Geom., to appear, arXiv:1605.09596.
  13. Dai S., Li Q., On cyclic Higgs bundles, Math. Ann., to appear, arXiv:1710.10725.
  14. Daskalopoulos G., Dostoglou S., Wentworth R., On the Morgan-Shalen compactification of the ${\rm SL}(2,{\bf C})$ character varieties of surface groups, Duke Math. J. 101 (2000), 189-207, arXiv:math.DG/9810034.
  15. Deroin B., Tholozan N., Dominating surface group representations by Fuchsian ones, Int. Math. Res. Not. 2016 (2016), 4145-4166, arXiv:1311.2919.
  16. Donaldson S.K., Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987), 127-131.
  17. Dumas D., Wolf M., Polynomial cubic differentials and convex polygons in the projective plane, Geom. Funct. Anal. 25 (2015), 1734-1798, arXiv:1407.8149.
  18. Eells Jr. J., Sampson J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160.
  19. Fathi A., Laudenbach F., Poénaru V., Thurston's work on surfaces, Mathematical Notes, Vol. 48, Princeton University Press, Princeton, NJ, 2012.
  20. Fredrickson L., Generic ends of the moduli space of ${\rm SL}(n,\mathbb C)$-Higgs bundles, arXiv:1810.01556.
  21. Fredrickson L., Perspectives on the asymptotic geometry of the Hitchin moduli space, SIGMA 15 (2019), 018, 20 pages, arXiv:1809.05735.
  22. Garcia-Prada O., Gothen P.B., Mundet i Riera I., The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations, arXiv:0909.448.
  23. Gothen P.B., Components of spaces of representations and stable triples, Topology 40 (2001), 823-850, arXiv:math.AG/9904114.
  24. Guest M.A., Harmonic maps, loop groups, and integrable systems, London Mathematical Society Student Texts, Vol. 38, Cambridge University Press, Cambridge, 1997.
  25. Guest M.A., Lin C.S., Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa, J. Reine Angew. Math. 689 (2014), 1-32, arXiv:1010.1889.
  26. Guichard O., An introduction to the differential geometry of flat bundles and of Higgs bundles, in The Geometry, Topology and Physics of Moduli Spaces of Higgs Bundles, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., Vol. 36, World Sci. Publ., Hackensack, NJ, 2018, 1-63.
  27. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  28. Hitchin N.J., Lie groups and Teichmüller space, Topology 31 (1992), 449-473.
  29. Jost J., Riemannian geometry and geometric analysis, 3rd ed., Universitext, Springer-Verlag, Berlin, 2002.
  30. Jost J., Partial differential equations, 2nd ed., Graduate Texts in Mathematics, Vol. 214, Springer, New York, 2007.
  31. Katzarkov L., Noll A., Pandit P., Simpson C., Harmonic maps to buildings and singular perturbation theory, Comm. Math. Phys. 336 (2015), 853-903, arXiv:1311.7101.
  32. Kobayashi S., Differential geometry of complex vector bundles,Publications of the Mathematical Society of Japan, Vol. 15, Princeton University Press, Princeton, NJ, 1987.
  33. Labourie F., Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006), 51-114, arXiv:math.DG/0401230.
  34. Labourie F., Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007), 1057-1099, arXiv:math.DG/0611250.
  35. 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.
  36. Labourie F., Cyclic surfaces and Hitchin components in rank 2, Ann. of Math. 185 (2017), 1-58, arXiv:1406.4637.
  37. Li Q., Harmonic maps for Hitchin representations, Geom. Funct. Anal. 29 (2019), 539-560, arXiv:1806.06884.
  38. Loftin J., Flat metrics, cubic differentials and limits of projective holonomies, Geom. Dedicata 128 (2007), 97-106, arXiv:math.DG/0611289.
  39. Loftin J., Survey on affine spheres, in Handbook of Geometric Analysis, No. 2, Adv. Lect. Math. (ALM), Vol. 13, Int. Press, Somerville, MA, 2010, 161-191, arXiv:0809.1186.
  40. Loftin J.C., Affine spheres and convex $\mathbb{RP}^n$-manifolds, Amer. J. Math. 123 (2001), 255-274.
  41. Mazzeo R., Swoboda J., Weiss H., Witt F., Ends of the moduli space of Higgs bundles, Duke Math. J. 165 (2016), 2227-2271, arXiv:1405.5765.
  42. Mochizuki T., Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces, J. Topol. 9 (2016), 1021-1073, arXiv:1508.05997.
  43. Morgan J.W., Shalen P.B., Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. 120 (1984), 401-476.
  44. Parreau A., Compactification d'espaces de représentations de groupes de type fini, Math. Z. 272 (2012), 51-86, arXiv:1003.1111.
  45. Reznikov A.G., Harmonic maps, hyperbolic cohomology and higher Milnor inequalities, Topology 32 (1993), 899-907.
  46. Sacks J., Uhlenbeck K., The existence of minimal immersions of $2$-spheres, Ann. of Math. 113 (1981), 1-24.
  47. Sacks J., Uhlenbeck K., Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982), 639-652.
  48. Sampson J.H., Some properties and applications of harmonic mappings, Ann. Sci. 'Ecole Norm. Sup. (4) 11 (1978), 211-228.
  49. Schoen R., Yau S.T., On univalent harmonic maps between surfaces, Invent. Math. 44 (1978), 265-278.
  50. Schoen R., Yau S.T., Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. 110 (1979), 127-142.
  51. Schoen R.M., The role of harmonic mappings in rigidity and deformation problems, in Complex Geometry (Osaka, 1990), Lecture Notes in Pure and Appl. Math., Vol. 143, Dekker, New York, 1993, 179-200.
  52. Simon U., Wang C.P., Local theory of affine $2$-spheres, in Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., Vol. 54, Amer. Math. Soc., Providence, RI, 1993, 585-598.
  53. Simpson C.T., Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918.
  54. Thurston W.P., On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 417-431.
  55. Wang C.P., Some examples of complete hyperbolic affine $2$-spheres in ${\mathbb R}^3$, in Global Differential Geometry and Global Analysis (Berlin, 1990), Lecture Notes in Math., Vol. 1481, Springer, Berlin, 1991, 271-280.
  56. Wentworth R.A., Higgs bundles and local systems on Riemann surfaces, in Geometry and Quantization of Moduli Spaces, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2016, 165-219, arXiv:1402.4203.
  57. Wolf M., The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), 449-479.

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