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Obstruction flatness of a strongly pseudoconvex hypersurface $\Sigma$ in acomplex manifold refers to the property that any (local) K\"ahler-Einsteinmetric on the pseudoconvex side of $\Sigma$, complete up to $\Sigma$, has apotential $-\log u$ such that $u$ is $C^\infty$-smooth up to $\Sigma$. Ingeneral, $u$ has only a finite degree of smoothness up to $\Sigma$. In thispaper, we study obstruction flatness of hypersurfaces $\Sigma$ that arise asunit circle bundles $S(L)$ of negative Hermitian line bundles $(L, h)$ overK\"ahler manifolds $(M, g).$ We prove that if $(M,g)$ has constant Riccieigenvalues, then $S(L)$ is obstruction flat. If, in addition, all theseeigenvalues are strictly less than one and $(M,g)$ is complete, then we showthat the corresponding disk bundle admits a complete K\"ahler-Einstein metric.Finally, we give a necessary and sufficient condition for obstruction flatnessof $S(L)$ when $(M, g)$ is a K\"ahler surface $(\dim M=2$) with constant scalarcurvature.

K\"ahler-Einstein metrics and obstruction flatness of circle bundles