Kahler-Einstein metrics on strictly pseudoconvex domains

Extending the results of Cheng and Yau it is shown that a strictly pseudoconvex domain in a complex manifold carries a complete Kahler-Einstein metric if and only if its canonical bundle is positive, i.e. admits an Hermitian connection with positive curvature. We consider the restricted case in which the CR structure on is normal. In this case M must be a domain in a resolution of the Sasaki cone over . We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a Kahler-Einstein manifold. We are able to mostly determine those normal CR three-manifolds which can be CR infinities. We give many examples of Kahler-Einstein strictly pseudoconvex manifolds on bundles and resolutions. In particular, the tubular neighborhood of the zero section of every negative holomorphic vector bundle on a compact complex manifold whose total space satisfies c (1) < 0 admits a complete Kahler-Einstein metric.