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

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 kφk, then ln K ≤ Ckφk, where K is the maximal dilatation of the minimal Lagrangian extension of φ to the hyperbolic plane.