Quasicircles and width of Jordan curves in CP1

Francesco Bonsante,Jean‐Marc Schlenker,Sara Maloni,Jeffrey Danciger

doi:10.1112/blms.12438

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.

Highlights

We study a notion of “width” for Jordan curves in CP1, paying special attention to the class of quasicircles
Given a Jordan curve C in CP1, let CH(C) denote the convex hull of C in the 3-dimensional hyperbolic space H3, namely the smallest closed convex set whose accumulation set at infinity is C
It might be useful to point out an analogy between the results presented here and the relation between quasicircles and minimal surfaces in hyperbolic geometry

Summary

Results and motivations

In [BS10], Bonsante-Schlenker define the width wAdS(C) of an acausal meridian C in terms of the timelike distances between points of the future boundary ∂+CH(C) of the convex hull and points of the past boundary ∂−CH(C). Bonsante-Schlenker [BS10, Theorem 1.12] characterize Einstein quasicircles as those for which the width is strictly less than the maximum possible. Given an acausal curve Γ ⊂ ∂AdS3, it always bounds a maximal surface with principal curvatures at most 1, and Γ is a quasicircle if and only if it bounds a maximal surface with principal curvatures uniformly less than 1 [BS10] This analogy suggests natural questions, for instance whether a quasicircle in ∂H3 with width less than an explicit constant (perhaps w0) bounds a minimal surface with principal curvatures less than 1

Width does not characterize quasicircles in CP1

Optimality orem B optimal?

Comparing the width and the boundary width