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

TL;DR

This paper proves the existence and uniqueness of maximal surfaces in certain anti-de Sitter 3-manifolds with particles and relates this to Teichmüller theory, establishing a unique minimal Lagrangian diffeomorphism between hyperbolic surfaces with cone singularities.

Contribution

It introduces a new existence and uniqueness result for maximal surfaces in AdS manifolds with particles and links it to Teichmüller theory through minimal Lagrangian diffeomorphisms.

Findings

Unique maximal surfaces exist in AdS convex GHM manifolds with particles for cone angles less than π.

A unique minimal Lagrangian diffeomorphism exists between hyperbolic surfaces with the same cone angles less than π.

The results connect geometric structures in AdS manifolds to Teichmüller theory.

Abstract

We prove the existence of a unique maximal surface in each anti-de Sitter (AdS) convex 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 Teichm\"uller 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 $\pi$.