Strictly pseudoconvex domains in C^2 with obstruction flat boundary

Abstract:  On a bounded strictly pseudoconvex domain in C^n, n>1, the smoothness of the Cheng-Yau solution to Fefferman's Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded simply connected domains in C^2 with connected boundary, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball.