Geometriae Dedicata

, Volume 126, Issue 1, pp 187–254

Minimal surfaces and particles in 3-manifolds

Original Paper

DOI: 10.1007/s10711-007-9132-1

Cite this article as:
Krasnov, K. & Schlenker, JM. Geom Dedicata (2007) 126: 187. doi:10.1007/s10711-007-9132-1


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.


Anti-de sitterCone-manifoldsQuasi-fuchsianHyperbolicMinimal surfaces

Mathematics Subject Classifications (2000)

Primary 53C42Secondary 53C50Secondary 53C80

Copyright information

© 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