  On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the
smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere
equation up to the boundary is obstructed by a local CR invariant of the
boundary. For a bounded strictly pseudoconvex domain $\Omega\subset
\mathbb{C}^2$ diffeomorphic to the ball, we prove that the global vanishing of
this obstruction implies biholomorphic equivalence to the unit ball, subject to
the existence of a holomorphic vector field satisfying a mild approximate
tangency condition along the boundary. In particular, by considering the Euler
vector 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 on
establishing an integral identity involving the CR curvature of $\partial
\Omega$ for any holomorphic vector field defined in a neighborhood of the
boundary. The notion of ambient holomorphic vector field along the CR boundary
generalizes naturally to the abstract setting, and the corresponding integral
identity still holds in the case of abstract CR $3$-manifolds.
