# zbMATH — the first resource for mathematics

Cyclic surfaces and Hitchin components in rank 2. (English) Zbl 1372.32021
Let $$G_0$$ be a split real adjoint simple Lie group and $$S(G_0)$$ the associated symmetric space (i.e., the space of Cartan involutions of $$G_0$$). We recall that, given a connected oriented closed surface $$\Sigma$$ of genus $$g\geq 2$$ and the corresponding character variety $$\text{Hom}(\pi_1(\Sigma), G_0)/G_0$$ of representations from $$\pi_1(\Sigma)$$ to $$G_0$$, in [Topology 31, No. 3, 449–473 (1992; Zbl 0769.32008)] N. J. Hitchin introduced the fundamental notion of Teichmüller component, now called Hitchin component, which is a special connected component $$\mathcal H(\Sigma, G_0)$$ of $$\text{Hom}(\pi_1(\Sigma), G_0)/G_0$$ that contains the connected component of $$\text{Hom}(\pi_1(\Sigma), \text{PSL}(2, \mathbb R))/\text{PSL}(2,\mathbb R)$$ corresponding to the Teichmüller space of $$\Sigma$$. We also recall that the elements of the Hitchin component $$\mathcal H(\Sigma, G_0)$$, the so-called Hitchin representations, are the monodromies of the flat connections $$\nabla + \Phi + \Phi^*$$ of some special Higgs bundles $$(\mathcal G, \Phi)$$ over $$\Sigma$$, in which $$\nabla$$ admits a real structure $$\rho$$ so that the pair $$(\nabla, \rho)$$ is solution to the self-duality equations of $$(\mathcal G, \Phi)$$. The special Higgs bundles $$(\mathcal G, \Phi)$$ to be considered are parametrised by families of holomorphic differentials $$q = (q_1, \ldots, q_\ell)$$, $$\ell = \text{rank}(G_0)$$, of degrees $$(m_1+1, \dots, m_\ell +1)$$, respectively, for some integers $$m_i$$ uniquely determined by the Lie group structure of $$G_0$$.
The main result of this paper is the following: When $$\text{rank}(G_0) = 2$$ (i.e., $$G_0 = \text{SL}(3, \mathbb R)$$, $$\text{PSp}(4, \mathbb R)$$ or $$\text{G}_{2,0}$$) for each Hitchin representation $$\delta \in \mathcal H(\Sigma, G_0)$$, there exists a uniquely associated $$\delta$$-equivariant minimal surface $$\phi^\delta: \Sigma \to S(G_0)$$ in the symmetric space $$S(G_0)$$ of $$G_0$$. Such map $$\phi^\delta$$ has the form $$\phi^\delta = p\circ f^\delta$$, where $$f^\delta: \Sigma \to \mathsf X$$ is an appropriate holomorphic map, called cyclic surface, into the space $$\mathsf X$$ of Hitchin-Kostant quadruples, and $$p: \mathsf X \to S(G_0)$$ is the natural projection of $$\mathsf X$$ onto $$S(G_0)$$.
Note that the existence part of this theorem was proved by the author in [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 3, 439–471 (2008; Zbl 1160.37021)] without any assumption on the rank. The uniqueness part is now reached by first proving a general criterion for infinitesimal rigidity of cyclic surfaces and then exploiting previous results of the author on the properness of the energy functionals on Teichmüller spaces determined by Hitchin representations.
Using the Hitchin parametrisation of the space of minimal surfaces, the above result also yields that for each $$G_0$$ as above, there exists an equivariant diffeomorphism between the Hitchin component $$\mathcal H(\Sigma, G_0)$$ and the space of pairs $$(J, Q)$$, given by a complex structure $$J$$ on $$\Sigma$$ and a $$J$$-holomorphic differential $$Q$$ of degree $$\frac{\dim(G_0) - 2}{2}$$.
Combining all this with the theory of positive bundles and the results of B. Berndtsson [Ann. Math. (2) 169, No. 2, 531–560 (2009; Zbl 1195.32012)], the author also obtains that for $$G_0$$ as above, the Hitchin component $$\mathcal H(\Sigma, G_0)$$ carries a complex structure and a $$1$$-dimensional family of compatible mapping class group invariant Kähler metrics, for which the Fuchsian locus is totally geodesic and whose restriction to the Fuchsian locus is the Weil-Petersson metric. The paper is then concluded with an Area Rigidity Theorem for Hitchin components.

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14H60 Vector bundles on curves and their moduli 32G13 Complex-analytic moduli problems 53C35 Differential geometry of symmetric spaces 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
##### Citations:
Zbl 0769.32008; Zbl 1160.37021; Zbl 1195.32012
Full Text:
##### References:
 [1] M. T. Anderson, ”Complete minimal hypersurfaces in hyperbolic $$n$$-manifolds,” Comment. Math. Helv., vol. 58, iss. 2, pp. 264-290, 1983. · Zbl 0549.53058 [2] M. F. Atiyah and R. Bott, ”The Yang-Mills equations over Riemann surfaces,” Philos. Trans. Roy. Soc. London Ser. A, vol. 308, iss. 1505, pp. 523-615, 1983. · Zbl 0509.14014 [3] D. Baraglia, ”Cyclic Higgs bundles and the affine Toda equations,” Geom. Dedicata, vol. 174, pp. 25-42, 2015. · Zbl 1321.53027 [4] Y. Benoist and D. Hulin, ”Cubic differentials and hyperbolic convex sets,” J. Differential Geom., vol. 98, iss. 1, pp. 1-19, 2014. · Zbl 1301.53040 [5] B. Berndtsson, ”Curvature of vector bundles associated to holomorphic fibrations,” Ann. of Math., vol. 169, iss. 2, pp. 531-560, 2009. · Zbl 1195.32012 [6] J. Bolton, F. Pedit, and L. Woodward, ”Minimal surfaces and the affine Toda field model,” J. Reine Angew. Math., vol. 459, pp. 119-150, 1995. · Zbl 0810.53048 [7] F. Bonsante and J. Schlenker, ”Maximal surfaces and the universal Teichmüller space,” Invent. Math., vol. 182, iss. 2, pp. 279-333, 2010. · Zbl 1222.53063 [8] N. Bourbaki, Lie Groups and Lie Algebras. Chapters 7-9, New York: Springer-Verlag, 2005. · Zbl 1139.17002 [9] S. B. Bradlow, O. Garc’ia-Prada, and P. B. Gothen, ”Deformations of maximal representations in $${ Sp}(4,\mathbb R)$$,” Q. J. Math., vol. 63, iss. 4, pp. 795-843, 2012. · Zbl 1261.14018 [10] M. Bridgeman, R. Canary, F. Labourie, and A. Sambarino, ”The pressure metric for Anosov representations,” Geom. Funct. Anal., vol. 25, iss. 4, pp. 1089-1179, 2015. · Zbl 1360.37078 [11] B. Collier and Q. Li, Asymptotics of certain families of Higgs bundles in the Hitchin component, 2014. · Zbl 1372.30046 [12] B. Collier and F. Labourie, Cyclic surfaces for products of $$\ms{SL}(2,\mathbb R)$$. [13] J. Demailly, Analytic Methods in Algebraic Geometry, Somerville, MA: International Press, 2012, vol. 1. · Zbl 1271.14001 [14] S. K. Donaldson, ”Twisted harmonic maps and the self-duality equations,” Proc. London Math. Soc., vol. 55, iss. 1, pp. 127-131, 1987. · Zbl 0634.53046 [15] D. Dumas and M. Wolf, ”Polynomial cubic differentials and convex polygons in the projective plane,” Geom. Funct. Anal., vol. 25, iss. 6, pp. 1734-1798, 2015. · Zbl 1335.30013 [16] V. Fock and A. Goncharov, ”Moduli spaces of local systems and higher Teichmüller theory,” Publ. Math. Inst. Hautes Études Sci., vol. 103, iss. 103, pp. 1-211, 2006. · Zbl 1099.14025 [17] O. García-Prada, P. B. Gothen, and I. Mundet i Riera, ”Higgs bundles and surface group representations in the real symplectic group,” J. Topol., vol. 6, pp. 64-118, 2013. · Zbl 1303.14043 [18] O. Garc’ia-Prada and I. Mundet i Riera, ”Representations of the fundamental group of a closed oriented surface in $${ Sp}(4,{\mathbb R})$$,” Topology, vol. 43, iss. 4, pp. 831-855, 2004. · Zbl 1070.14014 [19] W. M. Goldman, ”The symplectic nature of fundamental groups of surfaces,” Adv. in Math., vol. 54, iss. 2, pp. 200-225, 1984. · Zbl 0574.32032 [20] S. Choi and W. M. Goldman, ”Convex real projective structures on closed surfaces are closed,” Proc. Amer. Math. Soc., vol. 118, iss. 2, pp. 657-661, 1993. · Zbl 0810.57005 [21] P. A. Griffiths, ”Hermitian differential geometry and the theory of positive and ample holomorphic vector bundles,” J. Math. Mech., vol. 14, pp. 117-140, 1965. · Zbl 0134.39703 [22] O. Guichard and A. Wienhard, ”Convex foliated projective structures and the Hitchin component for $${ PSL}_4({\mathbf R})$$,” Duke Math. J., vol. 144, iss. 3, pp. 381-445, 2008. · Zbl 1148.57027 [23] O. Guichard and A. Wienhard, ”Anosov representations: domains of discontinuity and applications,” Invent. Math., vol. 190, iss. 2, pp. 357-438, 2012. · Zbl 1270.20049 [24] R. D. Gulliver II, R. Osserman, and H. L. Royden, ”A theory of branched immersions of surfaces,” Amer. J. Math., vol. 95, pp. 750-812, 1973. · Zbl 0295.53002 [25] N. Hitchin, ”The self-duality equations on a Riemann surface,” Proc. London Math. Soc., vol. 55, iss. 1, pp. 59-126, 1987. · Zbl 0634.53045 [26] N. Hitchin, ”Lie groups and Teichmüller space,” Topology, vol. 31, iss. 3, pp. 449-473, 1992. · Zbl 0769.32008 [27] Z. Huang and B. Wang, ”Counting minimal surfaces in quasi-Fuchsian three-manifolds,” Trans. Amer. Math. Soc., vol. 367, iss. 9, pp. 6063-6083, 2015. · Zbl 1322.53013 [28] A. Katok, ”Entropy and closed geodesics,” Ergodic Theory Dynam. Systems, vol. 2, iss. 3-4, pp. 339-365 (1983), 1982. · Zbl 0525.58027 [29] I. Kim and G. Zhang, Kähler metric on the space of convex real projective structures on surface. · Zbl 1373.57045 [30] S. Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton, NJ: Princeton Univ. Press, 1987, vol. 15. · Zbl 0708.53002 [31] B. Kostant, ”The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group,” Amer. J. Math., vol. 81, pp. 973-1032, 1959. · Zbl 0099.25603 [32] B. Kostant, ”Lie group representations on polynomial rings,” Amer. J. Math., vol. 85, pp. 327-404, 1963. · Zbl 0124.26802 [33] F. Labourie, ”Anosov flows, surface groups and curves in projective space,” Invent. Math., vol. 165, iss. 1, pp. 51-114, 2006. · Zbl 1103.32007 [34] F. Labourie, ”Flat projective structures on surfaces and cubic holomorphic differentials,” Pure Appl. Math. Q., vol. 3, iss. 4, part 1, pp. 1057-1099, 2007. · Zbl 1158.32006 [35] F. Labourie, ”Cross ratios, Anosov representations and the energy functional on Teichmüller space,” Ann. Sci. Éc. Norm. Supér., vol. 41, pp. 439-471, 2008. · Zbl 1160.37021 [36] F. cois Labourie and R. Wentworth, Variations along the Fuchsian locus, 2015. · Zbl 1404.37036 [37] Q. Li, ”Teichmüller space is totally geodesic in Goldman space,” Asian J. Math., vol. 20, iss. 1, pp. 21-46, 2016. · Zbl 1338.57019 [38] J. C. Loftin, ”Affine spheres and convex $$\mathbb{RP}^n$$-manifolds,” Amer. J. Math., vol. 123, iss. 2, pp. 255-274, 2001. · Zbl 0997.53010 [39] R. Potrie and A. Sambarino, Eigenvalues and entropy of a Hitchin representation, 2014. · Zbl 1380.30032 [40] J. Sacks and K. Uhlenbeck, ”The existence of minimal immersions of $$2$$-spheres,” Ann. of Math., vol. 113, iss. 1, pp. 1-24, 1981. · Zbl 0462.58014 [41] J. Sacks and K. Uhlenbeck, ”Minimal immersions of closed Riemann surfaces,” Trans. Amer. Math. Soc., vol. 271, iss. 2, pp. 639-652, 1982. · Zbl 0527.58008 [42] A. Sanders, Hitchin harmonic maps are immersions, 2014. [43] R. M. Schoen, ”The role of harmonic mappings in rigidity and deformation problems,” in Complex Geometry, Dekker, New York, 1993, pp. 179-200. · Zbl 0806.58013 [44] R. M. Schoen and S. T. Yau, ”Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature,” Ann. of Math., vol. 110, iss. 1, pp. 127-142, 1979. · Zbl 0431.53051 [45] C. T. Simpson, ”Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization,” J. Amer. Math. Soc., vol. 1, iss. 4, pp. 867-918, 1988. · Zbl 0669.58008 [46] M. Wolf, ”The Teichmüller theory of harmonic maps,” J. Differential Geom., vol. 29, iss. 2, pp. 449-479, 1989. · Zbl 0655.58009 [47] S. A. Wolpert, ”Chern forms and the Riemann tensor for the moduli space of curves,” Invent. Math., vol. 85, iss. 1, pp. 119-145, 1986. · Zbl 0595.32031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.