Classification of Holomorphic Mappings of Hyperquadrics from ${\mathbb{C}}^2$ to ${\mathbb{C}}^3$

We give a new proof of Faran’s and Lebl’s results by means of a new CR-geometric approach and classify all holomorphic mappings from the sphere in ${\mathbb{C}}^2$ to Levi-nondegenerate hyperquadrics in ${\mathbb{C}}^3$. We use the tools developed by Lamel, which allow us to isolate and study the most interesting class of holomorphic mappings. This family of so-called nondegenerate and transversal maps we denote by $\mathcal{F}$. For $\mathcal{F}$ we introduce a subclass $\mathcal{N}$ of maps which are normalized with respect to the group $\mathcal{G}$ of automorphisms fixing a given point. With the techniques introduced by Baouendi–Ebenfelt–Rothschild and Lamel we classify all maps in $\mathcal{N}$. This intermediate result is crucial to obtain a complete classification of $\mathcal{F}$ by considering the transitive part of the automorphism group of the hyperquadrics.