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

TL;DR

This paper investigates the curvature properties of constant mean curvature spacelike hypersurfaces in Anti-de Sitter space, introducing generalized convexity concepts and providing bounds on curvatures and quasiconformal maps.

Contribution

It generalizes convex hull notions and derives explicit curvature bounds based on the width of the convex hull in Anti-de Sitter space.

Findings

Bound on principal curvatures depending on convex hull width

Explicit bound on sectional curvature of H-hypersurfaces

Bound on quasiconformal dilatation of theta-landslides

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 $\theta-$landslides, in terms of the cross-ratio norm of their quasi-symmetric extension on $\partial_\infty\mathbb{H}^2$.