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 curvature 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.)