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.
Keywords
ACH metrics, the Einstein equation, self-duality, CR
manifolds, CR invariant differential operators
