Partial rigidity of CR embeddings of real hypersurfaces into hyperquadrics with small signature difference

We study the rigidity of holomorphic mappings from a neighborhood of a Levi-nondegenerate CR hypersurface M with signature l into a hyperquadric Q^N_l′ ⊆ ℂℙ^N+1 of larger dimension and signature. We show that if the CR complexity of M is not too large then the image of M under any such mapping is contained in a complex plane with a dimension depending only on the CR complexity and the signature difference, but not on N. This result follows from two theorems, the first demonstrating that for sufficiently degenerate mappings, the image of M is contained in a plane, and the second relating the degeneracy of mappings into different quadrics.