Bounded strictly pseudoconvex domains in C2 with obstruction flat boundary

On a bounded strictly pseudoconvex domain in $\Bbb{C}^n$, $n>1$, the smoothness of Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up boundary is obstructed by local curvature invariant boundary. For domains $\Bbb{C}^2$ which are diffeomorphic ball, we motivate and consider problem determining whether global vanishing this obstruction implies biholomorphic equivalence unit ball. In particular observe that, biholomorphism, ball rigid with respect deformations class flat We further show that for more general order equals CR curvature. Finally, give generalization recent result second author an abstract manifold transverse symmetry, flatness $3$-sphere.