A Burns-Epstein invariant for ACHE 4-manifolds

Abstract

We define a renormalized characteristic class for Einstein asymptotically complex hyperbolic (ache) manifolds of dimension 4: for any such manifold, the polynomial in the curvature associated to the characteristic class χ − 3τ is shown to converge. This extends a work of Burns and Epstein in the Kähler-Einstein case.We also define a new global invariant for any compact 3-dimensional pseudoconvex CR manifold, by a renormalization procedure of the η invariant of a sequence of metrics which approximate the CR structure.Finally, we get a formula relating the renormalized characteristic class to the topological number χ − 3τ and the invariant of the CR structure arising at infinity.1991 Mathematics Subject Classification. 53C55, 58J37, 58J60 (primary), 32V15, 58J28 (secondary).

Wormholes in ACH Einstein manifolds

“…In dimension 4, a much stronger asymptotic expansion is constructed in [3,Section 5], and one can take δ 0 = 2. In higher dimension, one must take only δ 0 = 1, because the Nijenhuis tensor of J is a first order invariant and occurs in the correction of g 0 at order 1.…”

Embeddability of some strongly pseudoconvex CR manifolds

“…Let X be a Sasakian manifold. In [BH02], O. Biquard and M. Herzlich construct a Kähler metric on the product (0, ∞) × X, called an asymptotically complex hyperbolic metric. X can then be viewed as the pseudoconvex boundary of some Kähler manifold, and therefore our technique can be applied to prove Theorem 1.4.…”

“…Asymptotically complex hyperbolic manifolds. In [BH02], O. Biquard and M. Herzlich consider a class of manifolds which are modelled on the complex unit ball, and are thus called asymptotically complex hyperbolic. This construction will allow us to get an embedding theorem for Sasakian manifolds, but let us first recall the meaning of an asymptotically complex hyperbolic manifold.…”

“…is then a metric on the contact distribution. Following Biquard and Herzlich [BH02], we endow Ω := (0, ∞) × X with the metric (5.1) g = dr 2 + e −2r θ 2 + e −r γ.…”

“…Actually, in [BH02], the authors consider the metric dr 2 + e 2r θ 2 + e r γ on Ω, but the reason for our choice will become clear later.…”