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

Andrea Seppi (1)

1 IF - Institut Fourier

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 homeo-morphism 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 ||φ||, then ln K ≤ C||φ||, where K is the maximal dilatation of the minimal Lagrangian extension of φ to the hyperbolic plane.

Domaines: Géométrie différentielle [math.DG]; Topologie géométrique [math.GT]

Andrea Seppi. Maximal surfaces in anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms. Journal of the European Mathematical Society, 2019, 21 (6), pp.1855-1913. ⟨10.4171/JEMS/875⟩. ⟨hal-01300369v3⟩