Bounded strictly pseudoconvex domains in $\mathbb{C}^2$ with obstruction flat boundary II

TL;DR

This paper proves that for certain strictly pseudoconvex domains in ^2, the vanishing of a boundary CR invariant ensures the domain is biholomorphic to the ball, under mild boundary conditions.

Contribution

It establishes a link between the vanishing of a CR invariant and biholomorphic equivalence to the ball, extending to abstract CR manifolds and involving integral identities with CR curvature.

Findings

Vanishing obstruction implies biholomorphic equivalence to the ball.

Integral identity involving CR curvature holds for holomorphic vector fields.

Results apply to a large neighborhood of the unit ball in ^2.

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-Ampere equation up to the boundary is obstructed by a local CR invariant of the boundary. For a bounded strictly pseudoconvex domain $\Omega\subset \mathbb{C}^2$ diffeomorphic to the ball, we prove that the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball, subject to the existence of a holomorphic vector field satisfying a mild approximate tangency condition along the boundary. In particular, by considering the Euler vector field multiplied by $i$ the result applies to all domains in a large $C^1$ open neighborhood of the unit ball in $\mathbb{C}^2$. The proof rests on establishing an integral identity involving the CR curvature of $\partial \Omega$ for any holomorphic vector field defined in a neighborhood of the boundary. The notion of ambient holomorphic vector field along the CR boundary generalizes naturally to the abstract setting, and the corresponding integral identity still holds in the case of abstract CR $3$-manifolds.