An improved bound for Sullivan's convex hull theorem

TL;DR

This paper improves the upper bound for the quasiconformal constant in Sullivan's convex hull theorem, providing a tighter estimate for conformally natural maps from hyperbolic domains to their convex hull boundaries.

Contribution

The authors establish a new upper bound of 7.1695 for the quasiconformal constant in Sullivan's theorem, refining previous bounds by Epstein, Marden, Markovic, and Bishop.

Findings

New explicit upper bound K_0 ≤ 7.1695 for Sullivan's convex hull theorem.

Improved bounds enhance understanding of conformal maps in hyperbolic geometry.

Refinement of previous estimates by notable mathematicians.

Abstract

Sullivan showed that there exists $K_0$ such that if $\Omega\subset \hat{\mathbb{C}}$ is a simply connected hyperbolic domain, then there exists a conformally natural $K_0$-quasiconformal map from $\Omega$ to the boundary ${\rm Dome}(\Omega)$ of the convex hull of its complement which extends to the identity on $\partial\Omega$. Explicit upper and lower bounds on $K_0$ were obtained by Epstein, Marden, Markovic and Bishop. We improve on these bounds, by showing that one may choose $K_0\le 7.1695$.