The Chains of Left-invariant CR-structures on SU(2)

TL;DR

This paper analyzes the chains of left-invariant CR structures on the three-sphere, revealing how their types vary with a real parameter and using geometric mechanics and Hamiltonian systems for the analysis.

Contribution

It provides a detailed classification of chains for these CR structures based on the parameter, including bifurcation analysis and integrable Hamiltonian systems.

Findings

Multiple chain types coexist for most parameters

A bifurcation occurs at a=√3 affecting chain types

The analysis uses a reduced Hamiltonian system similar to Euler's rigid body

Abstract

We compute the chains associated to the left-invariant CR structures on the three-sphere. These structures are characterized by a single real modulus $a$. For the standard structure $a=1$, the chains are well-known and are closed curves. We show that for almost all other values of the modulus $a$ either two or three types of chains are simultaneously present : (I) closed curves, (II) quasi-periodic curves dense on two-torii, or (III) chains homoclinic between closed curves. For $1 < a < \sqrt{3}$ no curves of the last type occur. A bifurcation occurs at $a = \sqrt{3}$ and from that point on all three types of chains are guaranteed to exist, and exhaust all chains. The method of proof is to use the Fefferman metric characterization of chains, combined with tools from geometric mechanics. The key to the computation is a reduced Hamiltonian system, similar to Euler's rigid body system, and depending on $a$, which is integrable.