Holomorphic mappings between hyperquadrics with small signature difference

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