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