Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics

TL;DR

This paper develops a framework for normal forms of Hermitian operators and applies it to classify CR maps between hyperquadrics and spheres, providing a comprehensive understanding of their structure and equivalence classes.

Contribution

It introduces a general approach to normal forms of Hermitian operators and applies it to classify CR maps between hyperquadrics and spheres, extending previous results.

Findings

Classified all real-analytic CR maps between hyperquadrics in $ ext{C}^2$ and $ ext{C}^3$

Proved all degree-two CR maps of spheres are spherically equivalent to monomial maps

Provided a finite list of equivalence classes for these CR maps

Abstract

We prove and organize some results on the normal forms of Hermitian operators composed with the Veronese map. We apply this general framework to prove two specific theorems in CR geometry. First, extending a theorem of Faran, we classify all real-analytic CR maps between any hyperquadric in $\C^2$ and any hyperquadric in $\C^3$, resulting in a finite list of equivalence classes. Second, we prove that all degree-two CR maps of spheres in all dimensions are spherically equivalent to a monomial map, thus obtaining an elegant classification of all degree-two CR sphere maps.