Quasicircles and width of Jordan curves in $\mathbb{CP}^1$

TL;DR

This paper introduces a geometric measure called 'width' for Jordan curves in the complex projective line, analyzing its relationship with quasicircles and demonstrating that small width implies quasicircle status.

Contribution

It defines and studies the width invariant for Jordan curves in P^1, establishing new criteria linking small width to being a quasicircle.

Findings

Jordan curves with small width are quasicircles

There exist Jordan curves of bounded width that are not quasicircles

The width invariant differs from the anti de Sitter setting in key ways

Abstract

We study a notion of "width" for Jordan curves in $\mathbb{CP}^1$, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schlenker to characterize quasicircles amongst a larger class of Jordan curves in the boundary of anti de Sitter space. By contrast to the AdS setting, we show that there are Jordan curves of bounded width which fail to be quasicircles. However, we show that Jordan curves with small width are quasicircles.