Generalized convexity and quantitative estimates for constant mean curvature spacelike hypersurfaces in Anti-de Sitter space

Abstract

We study the principal curvatures of properly embedded constant mean curvature hypersurfaces in the Anti-de Sitter space ${\mathbb{H}}^{n,1}$. We generalize the notion of convex hull and give an upper bound on the principal curvatures which only depends on the width of the $H-$shifted convex hull. This analysis has two direct consequences. First, it allows to bound the sectional curvature of $H-$hypersurfaces by an explicit function of the the width of the $H-$shifted convex hull. Second, we bound the quasiconfromal dilatation of a class of quasiconformal maps on the hyperbolic plane ${\mathbb{H}}^2$, called $\hat{I}¸-$landslides, in terms of the cross-ratio norm of their quasi-symmetric extension on ${\partial }_{\infty }{\mathbb{H}}^2$.

48 pages, 2 figures