Spherical CR uniformization of Dehn surgeries of the Whitehead link complement

TL;DR

This paper demonstrates how to produce infinitely many spherical CR structures on Dehn surgeries of the Whitehead link complement by deforming a known uniformization in complex hyperbolic space.

Contribution

It introduces a method to deform Ford domains in complex hyperbolic space to obtain new spherical CR structures on Dehn surgeries of the Whitehead link complement.

Findings

Infinitely many Dehn surgeries admit spherical CR structures.

Deformation of Ford domains yields new uniformizations.

Special case recovers the Figure Eight knot complement.

Abstract

We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the complex hyperbolic plane $\mathbb{H}^2_{\mathbb{C}}$. We deform the Ford domain of Parker and Will in $\mathbb{H}^2_{\mathbb{C}}$ in a one parameter family. On the one side, we obtain infinitely many spherical CR uniformizations on a particular Dehn surgery on one of the cusps of the Whitehead link complement. On the other side, we obtain spherical CR uniformizations for infinitely many Dehn surgeries on the same cusp of the Whitehead link complement. These manifolds are parametrized by an integer $n \geq 4$, and the spherical CR structure obtained for $n = 4$ is the Deraux-Falbel spherical CR uniformization of the Figure Eight knot complement.