Maximal surfaces in anti-de Sitter 3-manifolds with particles

Toulisse, Jérémy

Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1409-1449.

On démontre l’existence d’une unique surface maximale dans les variétés anti-de Sitter (AdS) globalement hyperboliques maximales (GHM) à particules (c’est à dire, avec des singularités coniques le long de courbes de type temps) lorsque les angles sont inférieurs à $\pi$. On interprète ce résultat en termes de théorie de Teichmüller et nous démontrons l’existence d’un unique difféomorphisme minimal lagrangien isotope à l’identité entre deux surfaces hyperboliques à singularités coniques, lorsque les angles singuliers sont les mêmes pour les deux surfaces et sont inférieurs à $\pi$.

We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) Globally Hyperbolic Maximal (GHM) manifold with particles (that is, with conical singularities along time-like lines) for cone angles less than $\pi$. We interpret this result in terms of Teichmüller theory, and prove the existence of a unique minimal Lagrangian diffeomorphism isotopic to the identity between two hyperbolic surfaces with cone singularities when the cone angles are the same for both surfaces and are less than $\pi$.

Classification : 53C42, 53C50
Keywords: maximal surfaces, cone-manifolds, Lorentz geometry, minimal Lagrangian maps
Mot clés : surfaces maximales, variétés à singularités coniques, géometrie lorentzienne, applications minimales lagrangiennes

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