Area-preserving diffeomorphism of the hyperbolic plane and K-surfaces in Anti-de Sitter space

TL;DR

This paper establishes a correspondence between certain spacelike surfaces in Anti-de Sitter space and area-preserving diffeomorphisms of the hyperbolic plane, leading to new results on the boundary behavior and extensions of quasisymmetric homeomorphisms.

Contribution

It introduces a representation formula linking spacelike surfaces in Anti-de Sitter space with area-preserving diffeomorphisms, and proves existence and uniqueness of K-surfaces with prescribed boundary curves.

Findings

Every weakly acausal boundary curve bounds two K-surfaces with specified convexity.

Quasisymmetric boundary curves correspond to K-surfaces with bounded principal curvatures.

Every quasisymmetric homeomorphism admits a unique quasiconformal extension as a hyperbolic landslide.

Abstract

We prove that any weakly acausal curve $\Gamma$ in the boundary of Anti-de Sitter (2+1)-space is the asymptotic boundary of two spacelike $K$-surfaces, one of which is past-convex and the other future-convex, for every $K\in(-\infty,-1)$. The curve $\Gamma$ is the graph of a quasisymmetric homeomorphism of the circle if and only if the $K$-surfaces have bounded principal curvatures. Moreover in this case a uniqueness result holds. The proofs rely on a well-known correspondence between spacelike surfaces in Anti-de Sitter space and area-preserving diffeomorphisms of the hyperbolic plane. In fact, an important ingredient is a representation formula, which reconstructs a spacelike surface from the associated area-preserving diffeomorphism. Using this correspondence we then deduce that, for any fixed $\theta\in(0,\pi)$, every quasisymmetric homeomorphism of the circle admits a unique extension which is a $\theta$-landslide of the hyperbolic plane. These extensions are quasiconformal.