The Fefferman-Szegő Sphericity Criterion in Complex Dimension Three

Abstract

We establish a Fefferman-Szegő characterization of local CR sphericity for smoothly bounded strongly pseudoconvex domains in complex dimension three. We derive the boundary expansion of the normalized determinant of the Fefferman-Szegő metric and prove that its second-order coefficient is a universal multiple of the squared Chern-Moser curvature. Hence, vanishing of the second-order deviation from the ball model is equivalent to local sphericity. A logarithmic stability theorem for the associated Monge-Ampère determinant controls the remainder and completes the dimension-three case.