A Burns-Epstein invariant for ACHE 4-manifolds

TL;DR

This paper introduces a new invariant for 4-dimensional ACHE manifolds and CR structures, extending previous work by defining renormalized characteristic classes and relating them to topological and CR invariants.

Contribution

It defines a renormalized characteristic class for ACHE 4-manifolds and introduces a global invariant for CR 3-manifolds, extending Burns and Epstein's work.

Findings

Convergence of the polynomial in curvature for the characteristic class

A new invariant for CR 3-manifolds via eta invariant renormalization

A formula linking the characteristic class to topological and CR invariants

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 euler-3signature is shown to converge. This extends a work of Burns and Epstein in the Kahler-Einstein case. This extends a work of Burns and Epstein in the Kahler-Einstein case. We also define a new global invariant for any 3-dimensional pseudoconvex CR manifold, by a renormalization procedure of the eta 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 euler-3signature and the invariant of the CR structure arising at infinity.