# Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms

### Andrea Seppi

Université Grenoble Alpes, Gières, France

## Abstract

We give upper bounds on the principal curvatures of a maximal surface of nonpositive curvature in three-dimensional Anti-de Sitter space, which only depend on the width of the convex hull of the surface. Moreover, given a quasisymmetric homeomorphism $ϕ$, we study the relation between the width of the convex hull of the graph of $ϕ$, as a curve in the boundary of infinity of Anti-de Sitter space, and the cross-ratio norm of $ϕ$.

As an application, we prove that if $ϕ$ is a quasisymmetric homeomorphism of $RP_{1}$ with cross-ratio norm $∣∣ϕ∣∣$, then ln $K≤C∣∣ϕ∣∣$, where $K$ is the maximal dilatation of the minimal Lagrangian extension of $ϕ$ to the hyperbolic plane.

## Cite this article

Andrea Seppi, Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms. J. Eur. Math. Soc. 21 (2019), no. 6, pp. 1855–1913

DOI 10.4171/JEMS/875