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

arXiv comment48 pages, 32 figures. Several typos corrected and improvements made. Part of the contents overlaps with Section 5 of arXiv:1411.3412v2. To appear in Journal of the European Mathematical Society