We adapt the work of Donaldson on the hyperkähler geometry of the space of almost Fuchsian manifolds to the deformation space of GHMC anti-de Sitter structures. 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 also relate these structures 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 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.
We study the asymptotic behaviour of the holonomy of the flat connection associated to SL(3,R) and Sp(4,R) cyclic Higgs bundles in the Hitchin component along rays of cubic and quartic differentials, including generic paths that go through zeros of the holomorphic differentials. We also describe the geometry of the limiting equivariant harmonic map to the asymptotic cone.
We define a pseudo-Riemannian metric and a symplectic form on the SL(3,R)-Hitchin component that are compatible with the natural complex structure coming from the identification of Hit(3) with the bundle of holomorphic cubic differentials over Teichmüller space. We show that this metric is negative definite on the fibres and restricts to a multiple of the Weil-Petersson metric on the Fuchsian locus, which, moreover, is totally geodesic.
We describe a one-parameter family of diffeomorphisms between the moduli space of polynomial quadratic differentials of degree k on the complex plane and pairs of ideal polygons with k+2 vertices in the hyperbolic plane.
Following work of Mirzakhani and a recent result of Sun Zhe, we give a recursive formula for the volume of the moduli space of regular globally hyperbolic maximal anti-de Sitter structures with loxodromic boundary monodromy.
We study the asymptotic geometry of the planar minimal surfaces with polynomial growth in the symmetric space SL(n,R)/SO(n) belonging to the Hitchin section. This generalizes our previous result concerning Sp(4,R)/U(2).
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 R^{4 }. 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.