On the Total CR Twist of Transversal Curves in the 3-Sphere

TL;DR

This paper studies the total CR twist functional on transversal curves in the 3-sphere, providing explicit integration methods, characterizing closed critical curves, and presenting evidence for infinitely many such closed curves.

Contribution

It introduces a procedure for explicit integration of critical curves and characterizes conditions for closed curves, advancing understanding of CR geometry in the 3-sphere.

Findings

Explicit integration method for critical curves

Characterization of closed critical curves

Evidence for infinitely many closed critical curves

Abstract

We investigate the total CR twist functional on transversal curves in the standard CR 3-sphere $\mathrm S^3 \subset \mathbb C^2$. The question of the integration by quadratures of the critical curves and the problem of existence and properties of closed critical curves are addressed. A procedure for the explicit integration of general critical curves is provided and a characterization of closed curves within a specific class of general critical curves is given. Experimental evidence of the existence of infinite countably many closed critical curves is provided.