A criterion for local embeddability of three-dimensional CR structures
Let \((M, D, J)\) be a \(3\)-dimensional CR manifold with a contact structure \(D\) and a smooth field of endomorphisms \(J\) such that \(J^2 = -\mathrm{id}\). The local embeddability problem asks whether there exists a local embedding \(\iota: M \hookrightarrow\mathbb{C}^2\) such that the CR structures \(\iota M\) induced by \(\iota\) and by \(\mathbb{C}^2\) coincide, and this problem is equivalent to finding two functionally independent CR functions \(f\) satisfying \(\bar \partial f =0\). It turns out [\textit{J.-P. Rosay}, Ann. Inst. Fourier 39, No. 3, 811--823 (1989; Zbl 0674.32008)] that the existence of nonconstant solutions to this equation is not sufficient to guarantee embeddability of the CR structure. A theorem by \textit{H. Jacobowitz} [Pac. J. Math. 127, No. 1, 91--101 (1987; Zbl 0583.32050)] provides a criterion for embeddability of CR structures in terms of the canonical bundle. In this paper, the authors introduce a family of Lorentzian metrics on a circle bundle over the CR manifold \(M\) which is more general than the conformal class of Fefferman metrics but more special than the family of Lorentzian metrics that admits a shearfree congruence, introduced by \textit{A. Trautman} [Classical Quantum Gravity 19, No. 2, R1--R10 (2002; Zbl 1008.53054)]. The authors call these metrics FRT metrics for Fefferman, Robinson and Trautman. The authors prove that a pseudoconvex CR structure \((M, D, J)\) is locally embeddable if and only if it amits an FRT metric on a circle bundle over \(M\) for which the complexified Ricci curvature vanishes on the \(\alpha\)-planes given by the \(\sqrt{-1}\) eigenspace of \(J\).
scientific article; zbMATH DE number 7041961
scientific article