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

TL;DR

This paper establishes bounds on the curvature of maximal surfaces in Anti-de Sitter space and explores the relationship between convex hull width and quasisymmetric homeomorphisms, linking geometric and conformal properties.

Contribution

It provides new bounds on principal curvatures based on convex hull width and relates the cross-ratio norm of boundary homeomorphisms to the maximal dilatation of their extensions.

Findings

Upper bounds on principal curvatures depend only on convex hull width.

Relation between convex hull width and cross-ratio norm of boundary homeomorphisms.

Bound on maximal dilatation in terms of cross-ratio norm.

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.