Flows and a tangency condition for embeddable CR structures in dimension 3

TL;DR

This paper investigates the preservation of fillability of 3D CR structures under geometric flows, establishing conditions for embeddability and deriving a tangency condition for fillable structures in complex space.

Contribution

It introduces a second order equation that ensures the preservation of fillability under certain geometric flows and derives a tangency condition for embeddable CR structures.

Findings

Fillability is preserved under flows satisfying a specific second order equation.

A tangency condition for the space of fillable CR structures is derived.

Application to the torsion and Cartan flows demonstrates the theoretical results.

Abstract

We study the fillability (or embeddability) of 3-dimensional $CR$ structures under the geometric flows. Suppose we can solve a certain second order equation for the geometric quantity associated to the flow. Then we prove that if the initial $CR$ structure is fillable, then it keeps having the same property as long as the flow has a solution. We discuss the situation for the torsion flow and the Cartan flow. In the second part, we show that the above mentioned second order operator is used to express a tangency condition for the space of all fillable or embeddable $CR$ structures at one embedded in $\mathbb{C}^{2}.$