Super-rigidity for CR embeddings of real hypersurfaces into hyperquadrics

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 $H_0(M)\subset Q^{N_0}_l$ but $H(U)\not\subset Q^{N_0}_l$. 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 $H(M)\subset Q^{N}_l$ and $H(U)\not\subset Q^{N_0}_l$ must be the standard linear embedding of $Q^{N_0}_l$ into $Q^N_l$ up to conjugation by automorphisms of $Q^{N_0}_l$ and $Q^N_l$