Autodual Einstein versus Kähler-Einstein (Q2571738)

It is known that any strictly pseudoconvex domain in \(\mathbb{C}^2\) carries a complete Kähler-Einstein metric, the Cheng-Yau metric with ``conformal infinity'' CR structure of the boundary, but not all CR structures on \(S^3\) arise in this way [\textit{S. Y. Cheng} and \textit{S. T. Yau}, Commun. Pure Appl. Math. 33, 507--544 (1980; Zbl 0506.53031); \textit{O. Biquard}, Métriques d'Einstein asymptotiquement symmétriques (Astérisque 265, Paris: Société Mathématique de France) (2000; Zbl 0967.53030)]. Here the structures with boundaries at infinity of complete selfdual Einstein metrics on \(S^3\) are studied. 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.