A spherical CR structure on a 3-manifold is a geometric structure modeled on the boundary at infinity of the complex hyperbolic plane, or in other words a (G,X)-structure with G=PU(2,1), X=S3. I will discuss spherical CR uniformizations, which are a special kind of spherical CR structure that arises by taking the manifold at infinity of a quotient of the ball by the action of a discrete group of isometries. I will explain how to construct some explicit uniformizations, including a 1-parameter family of (pairwise non-conjugate) spherical CR uniformizations of the figure eight knot complement.
