Related papers: Polyhedral surfaces in anti-de Sitter (2+1)-spacet…

We first prove that given a hyperbolic metric $h$ on a closed surface $S$, any flat metric on $S$ with negative singular curvatures isometrically embeds as a convex polyhedral Cauchy surface in a unique future-complete flat globally…

We prove that globally hyperbolic compact anti-de Sitter (2+1)-spacetimes with strictly convex spacelike boundary that is either smooth or polyhedral and whose holonomy is close to Fuchsian are determined by the induced metric on the…

Let $S$ be a closed surface of genus at least $2$, let $h$ be a smooth metric of curvature $K<-1$ on $S$, and let $h_0$ be a hyperbolic metric on $S$. We show that there exists a unique quasifuchsian AdS spacetime with left metric isotopic…

We prove that any metric of non-positive curvature in the sense of Alexandrov on a compact surface can be isometrically embedded as a convex spacelike Cauchy surface in a flat spacetime of dimension (2+1). The proof follows from polyhedral…

Given two Fuchsian representations $\rho_l$ and $\rho_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_\rho=(\mathbb{H}^2/\rho_l(\pi_1(S)))\times…

Let $S$ be a closed surface of hyperbolic type. We show that, for every pair $(g_+,g_-)$ of negatively curved metrics over $S$ there exists a unique GHMC Minkowski spacetime $X$ into which $(S,g_+)$ and $(S,g_-)$ isometrically embed as…

We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of…

A Fuchsian polyhedron in hyperbolic space is a polyhedral surface invariant under the action of a Fuchsian group of isometries (i.e. a group of isometries leaving globally invariant a totally geodesic surface, on which it acts cocompactly).…

It is shown that any two-dimensional spacetimes with compact Cauchy surfaces can be causally isomorphically imbedded into the two-dimensional Einstein's static universe. Also, it is shown that any two-dimensional globally hyperbolic…

We introduce the (2+1)-spacetimes with compact space of genus g and with r gravitating particles which arise by ``Minkowskian suspensions of flat or hyperbolic cone surfaces'', by ``distinguished deformations'' of hyperbolic suspensions and…

We show that any metric on $S^2$ with Gauss curvature $K \geq -\kappa$ admits a $C^{1,1}$-isometric embedding into the hyperbolic space with sectional curvature $-\kappa$. We also give a sufficient condition for a metric on $S^2$ to be…

We equip the space of Cauchy hypersurfaces in a globally hyperbolic spacetime with a natural Hausdorff-type metric and study its properties, in particular completeness and local compactness, for Lorentzian manifolds and in more general…

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a…

In this paper we study the flat (n+1)-spacetimes admitting a Cauchy surface diffeomorphic to a compact hyperbolic n-manifold. We show how to construct a canonical future complete one among all such spacetimes sharing the same holonomy. We…

Let S be a compact surface of genus >1, and g be a metric on S of constant curvature K\in\{-1,0,1\} with conical singularities of negative singular curvature. When K=1 we add the condition that the lengths of the contractible geodesics are…

We prove that for any convex globally hyperbolic maximal (GHM) anti-de Sitter (AdS) 3-dimensional space-time $N$ with particles (cone singularities of angles less than $\pi$ along time-like curves), the complement of the convex core in $N$…

Four-dimensional spacetimes foliated by a two-parameter family of homologous two-surfaces are considered in Einstein's theory of gravity. By combining a 1+(1+2) decomposition, the canonical form of the spacetime metric and a suitable…

We prove that any metric with curvature $\leq -1$ (in the sense of A. D. Alexandrov) on a closed surface of genus $>1$ is isometric to the induced intrinsic metric on a space-like convex surface in a Lorentzian manifold of dimension $(2+1)$…

We study the relationship between Lorentz harmonic maps into the hyperbolic plane and spacelike surfaces in anti-de Sitter 3-space. Using loop group techniques, we develop a DPW-type representation for Lorentz harmonic maps and provide an…

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…