SELF-DUAL EINSTEIN ACH METRICS AND CR GJMS OPERATORS IN DIMENSION THREE

摘要

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 and Graham. We also prove the convergence of the formal solutions when the CR structure is real analytic.