CR-structures on three dimensional circle bundles
In an earlier paper with \textit{D. M. Burns} [J. Am. Math. Soc. 3, No. 4, 809-841 (1990; Zbl 0736.32017)] the author considered the problem on embedding 3-dimensional \(CR\) manifolds in \(\mathbb{C}^ n\). The present paper along this line considers perturbations of \(U(1)\)-invariant \(CR\)- structures on circle bundles over surfaces. Let \(M\) be a contact manifold endowed with a \(U(1)\)-invariant strictly pseudoconvex \(CR\)-structure with the condition that \(M\) is diffeomorphic to a circle bundle over a compact surface (with a free transverse contact action by \(U(1))\). Let \(M/U(1)=\Sigma\). Then certain \(CR\) functions on \(M\) are in one to one correspondence with holomorphic sections of line bundles \(L_ n\) over \(\Sigma\). Moreover \(M\) is biholomorphic to a hypersurface in \(L_{-1}\). Under the assumption that \(\deg L_{-1}<0\) and others, it is shown here that certain perturbations of a \(U(1)\)-invariant \(CR\) structure are embeddable as small perturbations of certain embeddings. This type of theorem is considered first for those deformations with ``positive Fourier coefficients'' and for circle bundle of degree --1, and then for deformation of ``nonnegative Fourier coefficients'' and circle bundle of arbitrary degree. Another topic treated is the generic nonembeddability of real analytic \(U(1)\)-invariant structures. For 3-dimensional strictly pseudoconvex \(CR\) manifolds with trivial tangent bundle, an explicit obstruction to embedding is constructed, and the variational analysis of \(\square_ b\)- operator is considered. Finally the question of stability of embeddings is discussed. It is proved that if \(M\) is a circle bundle of degree \(-k\) over a compact Riemann surface of genus \(g\) with \(k \geq g\) \((k \geq 2g- 1)\), then any family \(\psi (t)\) of perturbations of \(U(1)\)-invariant structure with ``positive'' (``nonnegative'') Fourier coefficient is stably embeddable at \(t=0\).
scientific article; zbMATH DE number 220741
scientific article