A bounded strictly pseudoconvex domain in C^n, n>1, supports a
unique complete Kahler-Einstein metric determined by the Cheng-Yau solution
of Fefferman's Monge-Ampere equation. The smoothness of the solution of
Fefferman's equation up to the boundary is obstructed by a local CR
invariant of the boundary called the obstruction density. In the case n=2
the obstruction density is especially important, in particular in describing
the logarithmic singularity of the Bergman kernel. For domains in C^2
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. (This is a strong form of the
Ramadanov Conjecture.)