The moduli space of flat maximal space-like embeddings in pseudo-hyperbolic space

We study the moduli space of flat maximal space-like embeddings in $\mathbb{H}^{2,2}$ from various aspects. We first describe the associated Codazzi tensors to the embedding in the general setting, and then, we introduce a family of pseudo-K\"ahler metrics on the moduli space. We show the existence of two Hamiltonian actions with associated moment maps and use them to find a geometric global Darboux frame for any symplectic form in the above family.Comment: 31 page

Pseudo-K\"ahler structure on the SL(3,R)\mathrm{SL}(3,\mathbb{R})-Hitchin component and Goldman symplectic form

The aim of this paper is to show the existence and give an explicit description of a pseudo-Riemannian metric and a symplectic form on the $\mathrm{S}\mathrm{L}(3,\mathbb{R})$-Hitchin component, both compatible with Labourie and Loftin's complex structure. In particular, they give rise to a mapping class group invariant pseudo-K\"ahler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichm\"uller space. By comparing our symplectic form with Goldman's $\boldsymbol{\omega}_G$, we prove that the pair $(\boldsymbol{\omega}_G, \mathbf{I})$ cannot define a K\"ahler structure on the Hitchin component.Comment: Title and introduction changed. Added a result regarding Goldman symplectic for

Pseudo-Kähler geometry of Hitchin representations and convex projective structures

In this thesis we study the symplectic and pseudo-Riemannian geometry of the PSL(3,R)-Hitchin component associated with a closed orientable surface, using an approach coming from the theory of symplectic reduction in an infinite-dimensional context. In the case where the closed surface is homeomorphic to a torus, for each choice of a smooth real function with certain properties, we prove the existence of a pseudo-Kähler metric on the deformation space of properly convex projective structures. Moreover, we define a circle action and a SL(2,R)-action on the aforementioned space, which turn out to be Hamiltonian with respect to our symplectic form, and we give an explicit description of the moment maps. Then, we study the symplectic geometry of the deformation space as a completely integrable Hamiltonian system, and we find a geometric global Darboux frame for the symplectic form using the theory of complete Lagrangian fibrations. In the case of higher genus we define a mapping class group invariant pseudo-Kähler metric on the Hitchin component, by using a general construction of Donaldson. The complex structure is exactly the one coming from the identification with the holomorphic bundle of cubic differentials over Teichmüller space. In particular, we prove that Wang's equation for hyperbolic affine spheres has an interpretation as moment map for the action of an infinite-dimensional Lie group

Riemannian geometry of maximal surface group representations acting on pseudo-hyperbolic space

For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise to a non-degenerate Riemannian metric on the smooth locus of the subset consisting of maximal representations inside the character variety. In the case $n=2$, we carefully study the properties of the Riemannian metric on the maximal connected components, proving that it is compatible with the orbifold structure and finding some totally geodesic sub-varieties. Then, in the general case, we explain when a representation with Zariski closure contained in $\mathrm{SO}_0(2,3)$ represents a smooth or orbifold point in the maximal $\mathrm{SO}_0(2,n+1)$-character variety and we show that the associated space is totally geodesic for any $n\ge 3$.Comment: 38 page