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

TL;DR

This paper provides a new CR-geometric proof classifying all holomorphic mappings from the sphere in c^2 to Levi-nondegenerate hyperquadrics in c^3, extending previous results with novel techniques.

Contribution

It introduces a new CR-geometric approach and classifies all nondegenerate, transversal holomorphic maps between these hyperquadrics, refining prior classifications.

Findings

Complete classification of maps in the normalized subclass a0

Identification of the transitive automorphism group actions

Extension of Faran's and Lebl's results with new proof techniques

Abstract

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.