Maximal surfaces and the universal Teichm\"uller space

TL;DR

This paper establishes a correspondence between elements of the universal Teichmüller space and unique minimal Lagrangian diffeomorphisms of the hyperbolic plane, using maximal surfaces in anti-de Sitter space.

Contribution

It introduces a novel geometric approach linking maximal surfaces in AdS space to the structure of the universal Teichmüller space and characterizes quasi-symmetric homeomorphisms via 3D projective geometry.

Findings

Every element of the universal Teichmüller space corresponds to a unique minimal Lagrangian diffeomorphism.

Maximal space-like hypersurfaces in AdS space are bounded by prescribed boundary sets.

Quasi-symmetric homeomorphisms are characterized by maximal surfaces with negative curvature.

Abstract

We show that any element of the universal Teichm\"uller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in $AdS^{n+1}$, any subset $E$ of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In $AdS^3$, if $E$ is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.