Monotonicity for the Chern-Moser-Weyl curvature tensor and CR embeddings
\textit{S. M. Webster} [Duke Math. J. 45, No. 1, 39--46 (1978; Zbl 0373.32012)] showed that any nondegenerate hypersurface in \(\mathbb{C}^{n+1}\), having signature \(\ell\), can be embedded into the hyperquadric \(\mathbf{H}^{n+2}_{\ell+1}\subset\mathbb{C}^{n+2}\), of signature \(\ell+1\). Here the authors prove that, in general, there is no embedding in a quadric \(\mathbf{H}^{N+1}_{\ell}\subset\mathbb{C}^{N+1}\) of the same signature \(\ell\), even taking very large \(N\). This result, that does not apply to the pseucoconvex case (\(\ell=0\)) is obtained by proving a monotonicity result for the Chern-Moser-Weyl curvature tensor, under \(CR\) embeddings, along the complex directions that are isotropic for the Levi form.
scientific article; zbMATH DE number 5760408
scientific article