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 , we study the relation between the width of the convex hull of the graph of , as a curve in the boundary of infinity of Anti-de Sitter space, and the cross-ratio norm of .
As an application, we prove that if is a quasisymmetric homeomorphism of with cross-ratio norm , then ln , where is the maximal dilatation of the minimal Lagrangian extension of to the hyperbolic plane.
Cite this article
Andrea Seppi, Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms. J. Eur. Math. Soc. 21 (2019), no. 6, pp. 1855–1913DOI 10.4171/JEMS/875