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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.)
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