The aim of this chapter is to provide an introduction to Anti-de Sitter geometry, with special
emphasis on dimension three and on the relations with Teichmüller theory, whose study has …

Let $ M $ be a compact orientable 3-manifold with hyperbolizable interior and non-empty
boundary such that all boundary components have genii at least 2. We study an Alexandrov …

Celebrated work of Alexandrov and Pogorelov determines exactly which metrics on the
sphere are induced on the boundary of a compact convex subset of hyperbolic three-space …

Let be a closed surface of hyperbolic type. We show that, for every pair of negatively curved
metrics over, there exists a unique globally hyperbolic, maximal, and Cauchy compact …

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 …

We use Clifford algebras to construct a unified formalism for studying constant extrinsic
curvature immersed surfaces in Riemannian and semi-Riemannian 3-manifolds in terms of …

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 …

Let SS be a closed surface of genus at least 2, let hh be a smooth metric of curvature K<− 1
K<-1 on SS, and let h 0 h_0 be a hyperbolic metric on S S. We show that there exists a …

Let $ h^{+} $ and $ h^{-} $ be two complete, conformal metrics on the disc $\mathbb {D} $.
Assume moreover that the derivatives of the conformal factors of the metrics $ h^{+} $ and …

A compact Fuchsian manifold with boundary is a hyperbolic 3-manifold homeomorphic to
such that the boundary component is geodesic. We prove that a compact Fuchsian manifold …