Holomorphic Mappings between Hyperquadrics with Small Signature Difference

Abstract

In this paper, we study holomorphic mappings sending a hyperquadric of signature $\ell$ in $\bC^n$ into a hyperquadric of signature ${\ell }^{\prime }$ in $\bC^N$. We show (Theorem \ref{main}) that if the signature difference ${\ell }^{\prime }-\ell$ is not too large, then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and, in particular, the image of the mapping is contained in a complex plane of a dimension that depends only on $\ell$ and ${\ell }^{\prime }$, and not on the target dimension $N$. We also prove a Hopf Lemma type result (Theorem \ref{main2}) for such mappings.