Vol. 301, No. 2, 2019

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

Taiji Marugame

Vol. 301 (2019), No. 2, 519–546
DOI: 10.2140/pjm.2019.301.519

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.

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ACH metrics, the Einstein equation, self-duality, CR manifolds, CR invariant differential operators
Mathematical Subject Classification 2010
Primary: 32V05
Secondary: 53A55
Received: 11 February 2018
Revised: 25 November 2018
Accepted: 27 November 2018
Published: 24 October 2019
Taiji Marugame
Institute of Mathematics, Academia Sinica