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On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, thesmoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampereequation up to the boundary is obstructed by a local CR invariant of theboundary. For a bounded strictly pseudoconvex domain $\Omega\subset\mathbb{C}^2$ diffeomorphic to the ball, we prove that the global vanishing ofthis obstruction implies biholomorphic equivalence to the unit ball, subject tothe existence of a holomorphic vector field satisfying a mild approximatetangency condition along the boundary. In particular, by considering the Eulervector field multiplied by $i$ the result applies to all domains in a large$C^1$ open neighborhood of the unit ball in $\mathbb{C}^2$. The proof rests onestablishing an integral identity involving the CR curvature of $\partial\Omega$ for any holomorphic vector field defined in a neighborhood of theboundary. The notion of ambient holomorphic vector field along the CR boundarygeneralizes naturally to the abstract setting, and the corresponding integralidentity still holds in the case of abstract CR $3$-manifolds.

Bounded strictly pseudoconvex domains in $\mathbb{C}^2$ with obstruction flat boundary II