Kähler-Einstein metrics on strictly pseudoconvex domains

Abstract

The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly pseudoconvex domains in a complex manifold. Such a manifold carries a complete Kähler-Einstein metric if and only if its canonical bundle is positive. We consider the restricted case in which the CR structure on $\partial M$ is normal. In this case M must be a domain in a resolution of the Sasaki cone over $\partial M$. We give a condition on a normal CR manifold which it cannot satisfy if it is a CR infinity of a Kähler-Einstein manifold. We are able to mostly determine those normal CR 3-manifolds which can be CR infinities. Many examples are given of Kähler-Einstein strictly pseudoconvex manifolds on bundles and resolutions.

30 pages, 1 figure, couple corrections, improved a couple examples

References in corpus (5)

• Canonical Sasakian Metrics
• Rational singularities
• Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds
• Ricci-flat Kähler metrics on crepant resolutions of Kähler cones
• Examples of asymptotically conical Ricci-flat Kähler manifolds

Cited by in corpus (3)

• Q-prime curvature on CR manifolds
• Holomorphic family of strongly pseudoconvex domains in a Kähler manifold
• On the renormalized volume of tubes over polarized Kähler-Einstein manifolds