Geometriae Dedicata

, Volume 126, Issue 1, pp 187–254 | Cite as

Minimal surfaces and particles in 3-manifolds

Original Paper

Abstract

We consider 3-dimensional anti-de Sitter manifolds with conical singularities along time-like lines, which is what in the physics literature is known as manifolds with particles. We show that the space of such cone-manifolds is parametrized by the cotangent bundle of Teichmüller space, and that moreover such cone-manifolds have a canonical foliation by space-like surfaces. We extend these results to de Sitter and Minkowski cone-manifolds, as well as to some related “quasifuchsian” hyperbolic manifolds with conical singularities along infinite lines, in this later case under the condition that they contain a minimal surface with principal curvatures less than 1. In the hyperbolic case the space of such cone-manifolds turns out to be parametrized by an open subset in the cotangent bundle of Teichmüller space. For all settings, the symplectic form on the moduli space of 3-manifolds that comes from parameterization by the cotangent bundle of Teichmüller space is the same as the 3-dimensional gravity one. The proofs use minimal (or maximal, or CMC) surfaces, along with some results of Mess on AdS manifolds, which are recovered here in a different way, using differential-geometric methods and a result of Labourie on some mappings between hyperbolic surfaces, that allows an extension to cone-manifolds.

Keywords

Anti-de sitter Cone-manifolds Quasi-fuchsian Hyperbolic Minimal surfaces 

Mathematics Subject Classifications (2000)

Primary 53C42 Secondary 53C50 Secondary 53C80 

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© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of NottinghamNottinghamUK
  2. 2.Institut de Mathématiques, UMR CNRS 5219Université Paul SabatierToulouse Cedex 9France

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