Related papers: Regular globally hyperbolic maximal anti-de Sitter…
Let $\Sigma$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce wild globally hyperbolic anti-de Sitter structures on $\Sigma \times \mathbb{R}$ and provide two parameterisations of their…
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 $S \times \mathbb{R}$ by the cotangent bundle over the Teichm\"uller…
Using the parameterisation of the deformation space of GHMC anti-de Sitter structures on $S \times \mathbb{R}$ by the cotangent bundle of the Teichm\"uller space of $S$, we study how some geometric quantities, such as the Lorentzian…
Let $\Sigma$ be a surface of negative Euler characteristic, homeomorphic to a closed surface, possibly with a finite number of points removed. In this paper, we present a construction method for a wide range of examples of geometric…
A geometrical correspondence between maximal surfaces in anti-De Sitter space-time and minimal surfaces in the Riemannian product of the hyperbolic plane and the real line is established. New examples of maximal surfaces in anti-De Sitter…
In this paper we study the para-hyperK\"ahler geometry of the deformation space of MGHC anti-de Sitter structures on $\Sigma\times\mathbb R$, for $\Sigma$ a closed oriented surface. We show that a neutral pseudo-Riemannian metric and three…
We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $\pi$. We…
We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We…
We consider the Anti-de Sitter space $\mathbb{H}^3_1$ equipped with Berger-like metrics, that deform the standard metric of $\mathbb{H}^3_1$ in the direction of the hyperbolic Hopf vector field. Helix surfaces are the ones forming a…
We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these…
We prove that any weakly acausal curve $\Gamma$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$.…
In this paper 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…
De Sitter spacetime can be separated into two parts along two kinds of hypersurfaces and the half-de Sitter spacetimes are covered by the planar and hyperbolic coordinates respectively. Two positive energy theorems were proved previously…
Let $M$ be a globally hyperbolic maximal compact $3$-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…
We prove that any hyperbolic end with particles (cone singularities along infinite curves of angles less than $\pi$) admits a unique foliation by constant Gauss curvature surfaces. Using a form of duality between hyperbolic ends with…
A survey on the recent work of Danciger, Gu\'eritaud and Kassel on Margulis space-times and complete anti-de Sitter space-times. Margulis space-times are quotients of the 3-dimensional Minkowski space by (non-abelian) free groups acting…
In this paper, we introduce a new variation of the Teichm\"{u}ller space, namely the deformation space of hyperbolic structures on a surface with both enhancement and decoration. We construct the parameterization of this deformation space,…
We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space…
We provide the first examples of geometric transition from hyperbolic to anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic…
For oriented surfaces $\Sigma$ with boundary, we consider the infinite-dimensional deformation space of projective structures on $\Sigma$ with nondegenerate boundary, up to isotopies fixing the boundary. We show that this space carries a…