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

Source 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.