K\"{a}hler-Einstein metrics on strictly pseudoconvex domains

TL;DR

This paper investigates conditions under which strictly pseudoconvex domains in complex manifolds admit complete Kähler-Einstein metrics, focusing on cases with normal CR structures and providing classifications and examples.

Contribution

It characterizes when such domains have Kähler-Einstein metrics with normal CR boundaries and identifies which CR 3-manifolds can serve as CR infinities, with numerous examples.

Findings

A domain admits a Kähler-Einstein metric iff its canonical bundle is positive.

Normal CR structures impose restrictions on the existence of Kähler-Einstein metrics.

Many explicit examples of Kähler-Einstein manifolds are constructed on bundles and resolutions.

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 K\"{a}hler-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 K\"{a}hler-Einstein manifold. We are able to mostly determine those normal CR 3-manifolds which can be CR infinities. Many examples are given of K\"{a}hler-Einstein strictly pseudoconvex manifolds on bundles and resolutions.