Extension of \(CR\) structures on three dimensional compact pseudoconvex CR manifolds
The authors discuss the extension of CR structures for orientable abstract CR manifolds \(M\) of hypersurface type of real dimension \(3\). First they prove (Theorem 1.1) that if there exists a smooth real valued function \(\lambda\) on \(M\) which is strictly subharmonic at all points where the Levi form of \(M\) is zero, then \(M\) is a pseudo-concave component of the boundary \(\partial\Omega\) of a \(2\)-dimensional complex manifold with boundary. This extends a result of \textit{M. Christ} [Ann. Math. (2) 129, No. 1, 195--213 (1989; Zbl 0668.32019)] for an \(M\) of finite type. In case an extension to the \textit{pseudo-convex} side is already given, the assumptions of Theorem 1.1 yield an embedding of \(M\) into a \(2\)-dimensional complex manifold. Straightening the pseudo-convexity assumption for the CR structure, the authors obtain a local version of these results, that also applies to non-compact CR manifolds \(M\).
scientific article; zbMATH DE number 5013662