Holomorphic mappings between hyperquadrics with small signature difference

In this paper, we study holomorphic mappings sending a hyperquadric of signature $\ell$ in ${\Bbb C}^n$ into a hyperquadric of signature $\ell'$ in ${\Bbb C}^N$. We show (Theorem 1.1) that if the signature difference $\ell'-\ell$ 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 $\ell'$, and not on the target dimension $N$.\ We also prove a Hopf Lemma type result (Theorem 1.3) for such mappings.