Anti-de Sitter Geometry and Teichmüller Theory

Abstract

The aim of these notes 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 been initiated by the seminal paper of Geoffrey Mess in 1990. In the first part we give a broad introduction to Anti-de Sitter geometry in any dimension. The main results of Mess, including the classification of maximal globally hyperbolic Cauchy compact manifolds and the construction of the Gauss map, are treated in the second part. Finally, the third part contains related results which have been developed after the work of Mess, with the aim of giving an overview on the state-of-the-art.

How to construct harmonic maps to the hyperbolic plane

“…There is a similar analysis of the Wang equation, which is applied in the study of affine spheres in [5,18,23] and [16] explains the relation between the harmonic map and Wang equations with CMC surfaces and hyperbolic affine spheres in R 3 . Note finally that [27] extends [5] and explains the relation between harmonic map equation and anti-de Sitter geometry, see also [3].…”

Bending laminations on convex hulls of anti‐de Sitter quasi‐circles

“…The original source for most of the points described here is [21,Section 7], see also [1]. More details can be found in [12] or in the background sections of [5, 10, 11 15].…”

Parallel spinor flows on three-dimensional Cauchy hypersurfaces

“…Existence of a parallel spinor field is obstructed since it implies (M, g) to be a solution of Einstein equations with pure radiation type of energy momentum tensor [20]. This fact connects the study of parallel real spinors to the study of globally hyperbolic Lorentzian manifolds satisfying a given curvature condition, which has been a fundamental problem in global Lorentzian geometry since the seminal work of G. Mess [21], see [5] and references therein for more details.…”