we introduce complex minimal Lagrangian surfaces in the bi-complex hyperbolic space and study their relation with representations in SL(3,C). Our theory generalizes at the same time minimal Lagrangian surfaces in the complex hyperbolic plane, hyperbolic affine spheres in R³, and Bers embeddings the space of geodesics of H³. If these surfaces are equivariant under representations in SL(3,C), our approach generalizes the study of almost R-Fuchsian representations in SU(2,1), Hitchin representations in SL(3,R), and quasi-Fuchsian representations in SL(2,C). Moreover, we give a parameterization of SL(3,C)-quasi-Fuchsian representations by an open set in the product of two copies of the bundle of holomorphic cubic differentials over the Teichmüller space of S, from which we deduce that this space of representations is endowed with a bi-complex structure. In the process, we introduce bi-complex Higgs bundles as a new tool for studying representations into semisimple complex Lie groups.

We study the moduli space of flat maximal space-like embeddings in 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-Kähler 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.

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 PSL(3,R)-Hitchin component, both compatible with Labourie and Loftin’s complex structure. In particular, they give rise to a mapping class group invariant pseudo-Kähler structure on a neighborhood of the Fuchsian locus, which restricts to a multiple of the Weil-Petersson metric on Teichmüller space. Finally, generalizing a previous result in the case of the torus, we prove the existence of an Hamiltonian circle action.

In the Labourie-Loftin parametrization of the Hitchin component of surface group representations into SL(3,R), we prove an asymptotic formula for holonomy along rays in terms of local invariants of the holomorphic differential defining that ray. Globally, we show that the corresponding family of equivariant harmonic maps to a symmetric space converge to a harmonic map into the asymptotic cone of that space. The geometry of the image may also be described by that differential: it is weakly convex and a (one-third) translation surface. We define a compactification of the Hitchin component in this setting for triangle groups that respects the parametrization by Hitchin differentials.

In this paper we study a broad class of complete Hamiltonian integrable systems, namely the ones whose associated Lagrangian fibration is complete and has non compact fibers. By studying the associated complete Lagrangian fibration, we show that, under suitable assumptions, the integrals of motion can be taken as action coordinates for the Hamiltonian system. As an application we find global Darboux coordinates for a new family of symplectic forms defined on the deformation space of properly convex projective structures on the torus. In the last part of the paper, we deduce the expression for an arbitrary isometry of the space.

In this paper we prove the existence of a pseudo-Kähler structure on the deformation space of properly convex projective structures over the torus. In particular, the pseudo-Riemannian metric and the symplectic form are compatible with the complex structure inherited from the identification with the complement of the zero section of the total space of the bundle of cubic holomorphic differentials over the Teichmüller space. We show that the circle action given by rotation of the fibers is Hamiltonian and it preserves both the metric and the symplectic form. Finally, we prove the existence of a moment map for the SL(2,R)-action.

We show that, in the character variety of surface group representations into the Lie group PSL(2,R)×PSL(2,R), the compactification of the maximal component introduced by the second author is a closed ball upon which the mapping class group acts. We study the dynamics of this action. Finally, we describe the boundary points geometrically as (A1×A1,2)-valued mixed structures.

In this paper we study the para-hyperKähler geometry of the deformation space of MGHC anti-de Sitter structures on SxR, for S a closed oriented surface. We show that a neutral pseudo-Riemannian metric and three symplectic structures coexist with an integrable complex structure and two para-complex structures, satisfying the relations of para-quaternionic numbers. We show that these structures are directly related to the geometry of MGHC manifolds, via the Mess homeomorphism, the parameterization of Krasnov-Schlenker by the induced metric on K-surfaces, the identification with the cotangent bundle to Teichmüller space, and the circle action that arises from this identification. Finally, we study the relation to the natural para-complex geometry that the space inherits from being a component of the PSL(2,B)-character variety, where B is the algebra of para-complex numbers, and the symplectic geometry deriving from Goldman symplectic form.

We show that complete maximal surfaces in anti-de Sitter space, hyperbolic affine spheres in R³ and maximal surfaces in H^2,2 have finite total curvature if and only if they are conformally planar and their embedding data is determined by a holomorphic polynomial differential on the complex plane.

In this short note we describe a new interesting phenomenon about the Sp(4,R)-character variety. Precisely, we show that the Hitchin component and all Gothen components share the same boundary in our length spectrum compactification.

We find a compactification of the SO(2,3)-Hitchin component by studying the degeneration of the induced metric on the unique equivariant maximal surface in the 4-dimensonal pseudo-hyperbolic space of signature (2,2). In the process, we establish the closure in the space of projectivised geodesic currents of the space of flat metrics induced by holomorphic quartic differentials on a Riemann surface. As an application, we describe the behaviour of the entropy of Hitchin representations along rays of quartic differentials.

In this short note we explain how to adapt the construction of two Riemannian metrics on the SL(3,R)-Hitchin component to the deformation space of globally hyperbolic anti-de Sitter structures: the pressure metric and the Loftin metric (studied by Qiongling Li). We show that the former is degenerate and we characterize its degenerate locus, whereas the latter is nowhere degenerate and the Fuchsian locus is a totally geodesic copy of Teichmüller space endowed with a multiple of the Weil-Petersson metric.

We find a compactification of the SL(3,R)-Hitchin component by studying the degeneration of the Blaschke metrics on the associated equivariant affine spheres. In the process, we establish the closure in the space of projectivised geodesic currents of the space of flat metrics induced by holomorphic cubic differentials on a Riemann surface.

We combine our recent work on regular globally hyperbolic maximal anti-de Sitter structures with the classical theory of globally hyperbolic maximal Cauchy-compact anti-de Sitter manifolds in order to define an augmented moduli space of such structures. Moreover, we introduce a coordinate system in this space that resembles the complex Fenchel-Nielsen coordinates on hyperbolic quasi-Fuchsian manifolds.

We study the asymptotic geometry of a family of conformally planar minimal surfaces with polynomial growth in the Sp(4,R)-symmetric space. We describe a homeomorphism between the ``Hitchin component'' of wild Sp(4,R)-Higgs bundles over the Riemann sphere with a single pole at infinity and a component of maximal surfaces with light-like polygonal boundary in H^2,2. Moreover, we identify those surfaces with convex embeddings into the Grassmannian of symplectic planes of R⁴ . We show, in addition, that our planar maximal surfaces are the local limits of equivariant maximal surfaces in H^2,2 associated to Sp(4,R)-Hitchin representations along rays of holomorphic quartic differentials.

Let S be a connected, oriented surface with punctures and negative Euler characteristic. We define a family of globally hyperbolic maximal anti-de Sitter structures on S×I parameterised by the bundle over the Teichmüller space of S of meromorphic quadratic differentials on S with poles of order at least 3 at the punctures.

Let S be a closed oriented surface of genus at least 2. Using the parameterisation of the deformation space of globally hyperbolic maximal anti-de Sitter structures on S×I by the cotangent bundle over the Teichmüller space of S , we study the behaviour of these geometric structures along pinching sequences. We show, in particular, that regular globally hyperbolic anti-de Sitter structures naturally appear as limiting points.

Let S be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on S×I and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of S and as the bundle over Teichmüller space of meromorphic quadratic differentials with poles of order at most 2 at the punctures.

We construct geometrically a homeomorphism between the moduli space of polynomial quadratic differentials on the complex plane and light-like polygons in the 2-dimensional Einstein Universe. As an application, we find a class of minimal Lagrangian maps between ideal polygons in the hyperbolic plane.

Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on S×I by the cotangent bundle of the Teichmüller space of S, we study how some geometric quantities, such as the Lorentzian Hausdorff dimension of the limit set, the width of the convex core and the Hölder exponent, degenerate along rays of quadratic differentials.

Let M be a globally hyperbolic 3-dimensional spacetime locally modelled on Minkowski, Anti-de Sitter or de Sitter space. It is well known that M admits a unique foliation by constant mean curvature surfaces. In this paper we extend this result to singular spacetimes with particles (i.e. conical singularities of angle less than π along time-like geodesics).

We study the volume of maximal globally hyperbolic Anti-de Sitter manifolds containing a closed orientable Cauchy surface S , in relation to some geometric invariants depending only on the two points h and h' in Teichmüller space of S provided by Mess' parameterization. The main result of the paper is that the volume coarsely behaves like the minima of the L^1-energy of maps from (S,h) to (S,h'). A corollary of our result shows that the volume of maximal globally hyperbolic Anti-de Sitter manifolds is bounded from above by the exponential of (any of the two) Thurston's Lipschitz asymmetric distances, up to some explicit constants. Although there is no such bound from below, we provide examples in which this behavior is actually realized. We prove instead that the volume is bounded from below by the exponential of the Weil-Petersson distance.

We prove that, given an acausal curve in the boundary at infinity of Anti-de Sitter space which is the graph of a quasi-symmetric homeomorphism, there exists a foliation of its domain of dependence by constant mean curvature surfaces with bounded second fundamental form. Moreover, these surfaces provide a family of quasi-conformal extensions of the quasi-symmetric homeomorphism we started with.

We prove that given two metrics with curvature less than −1 on a closed, oriented surface of genus greater than 2, there exists an Anti-de-Sitter manifold with smooth, space-like, strictly convex boundary such that the induced metrics on the two connected components are equal to the two metrics we started with. Using the duality between convex space-like surfaces in Anti-de-Sitter space, we obtain an equivalent result about the prescription of the third fundamental form.