Obstruction Flat Rigidity of the CR 3-Sphere

Abstract

On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n >1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Amp\`ere equation up to the boundary is obstructed by a local curvature invariant of the boundary, the CR obstruction density $\mathcal{O}$. While local examples of obstruction flat CR manifolds are plentiful, the only known compact examples are the spherical CR manifolds. We consider the obstruction flatness problem for small deformations of the standard CR 3-sphere. That rigidity holds for the CR sphere was previously known (in all dimensions) for the case of embeddable CR structures, where it also holds at the infinitesimal level. In the 3-dimensional case, however, a CR structure need not be embeddable. While in the nonembeddable case we may no longer interpret the obstruction density $\mathcal{O}$ in terms of the boundary regularity of Fefferman's equation (or the logarithmic singularity of the Bergman kernel) the equation $\mathcal{O}\equiv 0$ is still of great interest, e.g., since it corresponds to the Bach flat equation of conformal gravity for the Fefferman space of the CR structure (a conformal Lorentzian 4-manifold). Unlike in the embeddable case, it turns out that in the nonembeddable case there is an infinite dimensional space of solutions to the linearized obstruction flatness equation on the standard CR 3-sphere and this space defines a natural complement to the tangent space of the embeddable deformations. In spite of this, we show that the CR 3-sphere does not admit nontrivial obstruction flat deformations, embeddable or nonembeddable.