Autodual Einstein Versus Kähler–Einstein

Abstract.

Any strictly pseudoconvex domain in $\mathbb{C}^2$ carries a complete Kähler-Einstein metric, the Cheng–Yau metric, with “conformal infinity” the CR structure of the boundary.

It is well known that not all CR structures on S³ arise in this way. In this paper, we study CR structures on the 3-sphere satisfying a different filling condition: boundaries at infinity of (complete) selfdual Einstein metrics. We prove that (modulo contactomorphisms) they form an infinite dimensional manifold, transverse to the space of CR structures which are boundaries of complex domains (and therefore of Kähler–Einstein metrics).