Constant mean curvature foliation of domains of dependence in $AdS_{3}$

TL;DR

This paper proves the existence and uniqueness of a foliation by constant mean curvature surfaces in the domain of dependence of an acausal boundary curve in AdS3, extending boundary homeomorphisms.

Contribution

It establishes a unique foliation of the domain of dependence in AdS3 by CMC surfaces associated with boundary curves, linking geometric and conformal boundary data.

Findings

Existence of a unique CMC foliation in $D( ext{boundary})$

These surfaces extend boundary homeomorphisms quasi-conformally

Provides a geometric framework connecting boundary curves and interior foliations

Abstract

We prove that, given an acausal curve $\Gamma$ in the boundary at infinity of $AdS_{3}$ which is the graph of a quasi-symmetric homeomorphism $\phi$, there exists a unique foliation of its domain of dependence $D(\Gamma)$ by constant mean curvature surfaces with bounded second fundamental form. Moreover, these surfaces provide a family of quasi-conformal extensions of $\phi$.