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

TL;DR

This paper investigates the geometric properties of bounded strictly pseudoconvex domains in ^2, focusing on the role of boundary invariants and their relation to biholomorphic equivalence to the unit ball, revealing rigidity and classification results.

Contribution

It establishes the rigidity of the unit ball under obstruction flatness and relates the vanishing order of the obstruction to CR curvature, extending previous results to more general settings.

Findings

Obstruction flatness implies biholomorphic equivalence to the unit ball in certain cases.

The order of vanishing of the obstruction matches the order of CR curvature vanishing.

Generalization of CR sphere equivalence under transverse symmetry.

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 curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\mathbb{C}^2$ which are 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. In particular we observe that, up to biholomorphism, the unit ball in $\mathbb{C}^2$ is rigid with respect to deformations in the class of strictly pseudoconvex domains with obstruction flat boundary. We further show that for more general deformations of the unit ball, the order of vanishing of the obstruction equals the order of vanishing of the CR curvature. Finally, we give a generalization of the recent result of the second author that for an abstract CR manifold with transverse symmetry, obstruction flatness implies local equivalence to the CR $3$-sphere.