HOLOMORPHIC MAPPINGS BETWEEN HYPERQUADRICS WITH SMALL SIGNATURE DIFFERENCE

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