We prove that regular GHM anti-de Sitter structures naturally appear as limits of GHMC manifolds along pinching sequences.
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.