Bending laminations on convex hulls of anti-de Sitter quasicircles

TL;DR

This paper proves the existence of fixed points for compositions of left earthquakes on the universal Teichmüller space using anti-de Sitter geometry, linking measured laminations to convex hull boundaries in AdS3.

Contribution

It establishes that any strongly filling pair of bounded measured laminations can be realized as bending laminations of convex hull boundaries in anti-de Sitter 3-space.

Findings

Fixed points exist for compositions of left earthquakes on Teichmüller space.

Any strongly filling pair of laminations can be realized via AdS convex hull boundaries.

The approach uses anti-de Sitter geometry to connect laminations with geometric structures.

Abstract

Let $\lambda_-$ and $\lambda_+$ be two bounded measured laminations on the hyperbolic disk $\mathbb H^2$, which "strongly fill" (definition below). We consider the left earthquakes along $\lambda_-$ and $\lambda_+$, considered as maps from the universal Teichm\"uller space $\mathcal T$ to itself, and we prove that the composition of those left earthquakes has a fixed point. The proof uses anti-de Sitter geometry. Given a quasi-symmetric homeomorphism $u:{\mathbb RP}^1\to {\mathbb RP}^1$, the boundary of the convex hull in $AdS^3$ of its graph in ${\mathbb RP}^1\times{\mathbb RP}^1\simeq \partial AdS^3$ is the disjoint union of two embedded copies of the hyperbolic plane, pleated along measured geodesic laminations. Our main result is that any pair of bounded measured laminations that "strongly fill" can be obtained in this manner.