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

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 $\mathbb{C}^{2}.$

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