Kähler–Einstein metrics on strictly pseudoconvex domains

Abstract

Extending the results of S. Y. Cheng and S.-T. Yau it is shown that a strictly pseudoconvex domain M ⊂ X in a complex manifold carries a complete Kähler-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 ∂M is normal. In this case M must be a domain in a resolution of the Sasaki cone over ∂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.We give many examples of Kähler-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 Kähler-Einstein metric.

On the Renormalized Volume of Tubes Over Polarized Kähler–Einstein Manifolds

“…Note that the assumption β < 1 is a necessary and sufficient condition for the existence of a complete Kähler-Einstein metric on Ω with negative Einstein constant, which has been proved by van Coevering [13].…”

Holomorphic Families of Strongly Pseudoconvex Domains in a Kähler Manifold

“…If there exists a Kähler form ω X on X and satisfies that the Ricci curvature Ric(ω y ) of ω y := ω X | Xy is negatively curved for every y ∈ S ′ , then Cheng and Yau's theorem implies that there exists a unique complete Kähler metric ω KE y on D y satisfying Ric(ω KE y ) = −(n + 1)ω KE y , where n is the dimension of D y (cf. [3,7]). This metric ω KE y is called the Kähler-Einstein metric with Ricci curvature −(n + 1).…”

“…Then a careful estimate of the boundary behavior of the geodesic curvature (which is a invariant encoding the positivity of a variation) completes the proof of Theorem 1.1. This is obtained by combining results and techniques in [4,5] and [7].…”

“…Lemma 3.4 in[7]). There exists a new defining function ψ = η · ψ of Ω with a positive function η ∈ C ∞ (Ω) such thatF = O(ψ n+1 ), where F := log (− ψ) −(n+1) (ω) n /(ω 0 ψ ) n .…”

Szegő kernel expansion and equivariant embedding of CR manifolds with circle action

“…From this result, we prove that if L is positive then the R-action comes from a torus action on X and by using the torus action, it is not difficult to show that if L is positive, then Spec (−iT ) is countable and any element in Spec (−iT ) is an eigenvalue of −iT . It was known before that the automorphism group of a compact Sasakian manifold is compact (see [14] and [3]) and therefore that the R-action induced by the Reeb flow comes from a torus action (see for example [2]). Using that result, an R-equivariant embedding result for compact Sasakian manifold (and hence for compact strongly pseudoconvex maniflolds with transversal CR vector fields) with vanishing first cohomology was proven in [2].…”