Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

M. S. Baouendi; P. Ebenfelt; X. Huang

paper

Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

arXiv:0711.4647

Abstract

Let $Q^N_l\subset \bC\bP^{N+1}$ denote the standard real, nondegenerate hyperquadric of signature $l$ and $M\subset \bC^{n+1}$ a real, Levi nondegenerate hypersurface of the same signature $l$ . We shall assume that there is a holomorphic mapping $H_0\colon U\to \bC\bP^{N_0+1}$, where $U$ is some neighborhood of $M$ in $\bC^{n+1}$, such that $H_{0} (M) \subset Q_{l}^{N_{0}}$ but $H (U) \neq \subset Q_{l}^{N_{0}}$ . We show that if $N_{0} - n < l$ then, for any $N \geq N_{0}$ , any holomorphic mapping $H\colon U\to \bC\bP^{N+1}$ with $H (M) \subset Q_{l}^{N}$ and $H (U) \neq \subset Q_{l}^{N_{0}}$ must be the standard linear embedding of $Q_{l}^{N_{0}}$ into $Q_{l}^{N}$ up to conjugation by automorphisms of $Q_{l}^{N_{0}}$ and $Q_{l}^{N}$ .