Self-dual Einstein ACH metrics and CR GJMS operators in dimension three

TL;DR

This paper constructs a family of Einstein ACH metrics with a given 3D CR structure, demonstrating self-duality and providing new proofs for CR invariant operators, with convergence results in the real analytic case.

Contribution

It refines Matsumoto's construction to produce ACH metrics solving Einstein equations to infinite order, and offers a new proof for the existence of CR GJMS operators of all orders.

Findings

Constructed a one-parameter family of ACH metrics solving Einstein equations to infinite order.

Proved the convergence of formal solutions for real analytic CR structures.

Provided an alternative proof for the existence of CR GJMS operators of all orders.

Abstract

By refining Matsumoto's construction of Einstein ACH metrics, we construct a one parameter family of ACH metrics which solve the Einstein equation to infinite order and have a given three dimensional CR structure at infinity. When the parameter is 0, the metric is self-dual to infinite order. As an application, we give another proof of the fact that three dimensional CR manifolds admit CR invariant powers of the sublaplacian (CR GJMS operators) of all orders, which has been proved by Gover-Graham. We also prove the convergence of the formal solutions when the CR structure is real analytic.