Convex hulls of quasicircles in hyperbolic and anti-de Sitter space.

Sara Maloni (University of Virginia)

Thurston conjectured that quasi-Fuchsian manifolds are uniquely determined by the induced hyperbolic metrics on the boundary of their convex core and Mess extended this conjecture to the context of globally hyperbolic anti de-Sitter spacetimes. In this talk I will discuss a universal version of Thurston and Mess’ conjectures: any quasisymmetric homeomorphism from the circle to itself is realized as the gluing map between the upper and lower boundary of the convex hull of quasicircle in the boundary at infinity of the 3-dimensional hyperbolic (resp. anti-de Sitter) space. We will also discuss a similar result for convex domains bounded by surfaces of constant curvature K in (−1, 0) in the hyperbolic setting and of curvature K < −1 in the anti de-Sitter setting with a quasicircle as their asymptotic boundary. (This is joint work in progress with F. Bonsante, J. Danciger and J.-M. Schlenker.)