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

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.Comment: 30 pages, 1 figure, couple corrections, improved a couple example

Null-geodesics in complex conformal manifolds and the LeBrun correspondence

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally flat. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal infinity of a seldual four-manifolds. We also find a relation between the conformal invariants of the conformal infinity and its ambient. (orig.)SIGLEAvailable from TIB Hannover: RR 1596(453) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

The Einstein-Dirac equation on Sasakian 3-manifolds

SIGLEAvailable from TIB Hannover: RR 1596(477) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman