In this paper, we study the geometric and dynamical properties of maximal representations of surface groups into Hermitian Lie groups of rank 2. Combining tools from Higgs bundle theory, the theory of Anosov representations, and pseudo-Riemannian geometry, we obtain various results of interest.We prove that these representations are holonomies of certain geometric structures, recovering results of Guichard and Wienhard. We also prove that their length spectrum is uniformly bigger than that of a suitably chosen Fuchsian representation, extending a previous work of the second author. Finally, we show that these representations preserve a unique minimal surface in the symmetric space, extending a theorem of Labourie for Hitchin representations in rank 2.

In this paper we describe the space of maximal components of the character variety of surface group representations into PSp(4, R) and Sp(4, R).For every real rank 2 Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups PSp(4, R) and Sp(4, R), we give a mapping class group invariant parameterization of each maximal component as an explicit holomorphic fiber bundle over Teichmüller space. Special attention is put on the connected components which are singular, we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components of PSp(4, R) and Sp(4, R) by the action of the mapping class group as a holomorphic submersion over the moduli space of curves.These results are proven in two steps, first we use Higgs bundles to give a non-mapping class group equivariant parameterization, then we prove an analogue of Labourie's conjecture for maximal PSp(4, R) representations.The spaces X max,sw2 sw1 (Γ, PSp(4, R)) are singular, but the singularities consist only of Z 2 and Z 2 ⊕ Z 2 -orbifold points. The space H ′ /Z 2 also has an orbifold structure. The homeomorphism above is an orbifold isomorphism, in particular, it is smooth away from the singular set.Corollary 7. Each space X max,sw2 sw1 (Γ, PSp(4, R)) deformation retracts onto the quotient of (S 1 ) 2g−2 by inversion. In particular, its rational cohomology is: PSp(4, R)), Q) ∼ = H j ((S 1 ) 2g−2 , Q) if j is even, 0 otherwise. Character varieties and Higgs bundlesIn this section we recall general facts about character varieties and Higgs bundles.

Abstract. Using Hitchin's parameterization of the Hitchin-Teichmüller component of the SL(n, R) representation variety, we study the asymptotics of certain families of representations. In fact, for certain Higgs bundles in the SL(n, R)-Hitchin component, we study the asymptotics of the Hermitian metric solving the Hitchin equations. This analysis is used to estimate the asymptotics of the corresponding family of flat connections as we scale the differentials by a real parameter. We consider Higgs fields that have only one holomorphic differential qn of degree n or q n−1 of degree (n − 1). The asymptotics of the corresponding parallel transport operator is calculated, and used to prove a special case of a conjecture of Katzarkov, Knoll, Pandit and Simpson [KNPS13] on the Hitchin WKB problem.

The Bia lynicki-Birula decomposition of the space of λ-connections restricts to the Morse stratification on the moduli space of Higgs bundles and to the partial oper stratification on the de Rham moduli space of holomorphic connections. For both the Morse and partial oper stratifications, every stratum is a holomorphic Lagrangian fibration over a component of the space of complex variations of Hodge structure. In this paper, we generalize known results for the Hitchin section and the space of opers to arbitrary strata. These include the following: a biholomorphic identification of the fibers of the two strata over a stable variation of Hodge structure via the " -conformal limit" of Gaiotto, a proof that the fibers of the Morse and partial oper stratifications are transverse at the base point, and an explicit parametrization of the fibers as half-dimensional affine spaces, 1

Let S be a closed surface of genus at least 2. For each maximal representation ρ : π 1 (S)→Sp(4, R) in one of the 2g − 3 exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space Sp(4, R)/U(2) is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmüller space. Unlike Labourie's recent results on Hitchin components, these bundles are not vector bundles.