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 among a larger class of Jordan curves in the boundary of anti de Sitter space. In 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.

中文翻译：

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CP1中的拟圆和约旦曲线的宽度

我们研究了约旦曲线的“宽度”概念 ${\mathbb{CP}}^{\mathrm{1个}}$，特别注意准圆的类别。约旦曲线的宽度是根据其双曲线三空间中的凸包的几何形状定义的。Bonsante–Schlenker使用了anti-de Sitter几何设置中的类似不变性来刻画anti de Sitter空间边界中较大乔丹曲线之间的拟圆。与AdS设置相比，我们显示存在宽度受限的约旦曲线，但这些曲线不是准圆。但是，我们显示出宽度较小的乔丹曲线为准圆。