Ambient and Poincaré metrics for CR $3$-manifolds associated with the self-dual Einstein ACH metric

We construct new ambient and Poincaré metrics for the Fefferman conformal manifold over $3$-dimensional CR manifolds, starting from the associated self-dual Einstein ACH metric. Unlike the classical ambient and Poincaré metrics arising from approximate solutions to the complex Monge-Ampère equation, these metrics satisfy Einstein-Maxwell-type equations to infinite order rather than the Ricci-flat or Einstein equations. Since these metrics are determined to infinite order without ambiguity, they enable us to construct CR invariant differential operators and local CR invariants involving arbitrarily high order derivatives of the Tanaka-Webster curvature and torsion. As an application, we obtain the ambient metric construction of the CR GJMS (Gover-Graham) operators of all orders in dimension $3$.