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

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 $\phi$, we study the relation between the width of the convex hull of the graph of $\phi$, as a curve in the boundary of infinity of Anti-de Sitter space, and the cross-ratio norm of $\phi$. As an application, we prove that if $\phi$ is a quasisymmetric homeomorphism of $\mathbb{R}\mathrm{P}^1$ with cross-ratio norm $||\phi||$, then $\ln K\leq C||\phi||$, where $K$ is the maximal dilatation of the minimal Lagrangian extension of $\phi$ to the hyperbolic plane.