Asymptotics of ACH-Einstein metrics

TL;DR

This paper investigates the boundary behavior of ACH Einstein metrics, revealing the existence of formal solutions with logarithmic terms and introducing a new CR invariant tensor that obstructs log-free solutions.

Contribution

It establishes the existence and uniqueness of formal asymptotic expansions for ACH Einstein metrics and introduces the CR obstruction tensor as a new invariant.

Findings

Existence of formal solutions with logarithmic terms.

Introduction of the CR obstruction tensor.

Properties of the CR invariant tensor discussed.

Abstract

We study the boundary asymptotics of ACH metrics which are formally Einstein. In terms of the partially integrable almost CR structure induced on the boundary at infinity, existence and uniqueness of such formal asymptotic expansions are studied. It is shown that there always exist formal solutions to the Einstein equation if we allow logarithmic terms, and that a local CR-invariant tensor arises as the obstruction to the existence of a log-free solution. Some properties of this new CR invariant, the CR obstruction tensor, are discussed.