2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63288We study the fillability (or embeddability) of $CR$ structures under the gauge-fixed Cartan flow. We prove that if the initial $CR$ structure is fillable with nowhere vanishing Tanaka-Webster curvature and free torsion, then it keeps having the same property after a short time. In the Appendix, we show the uniqueness of the solution to the gauge-fixed Cartan flow.17 pagesDifferential GeometryComplex Variables32G07, 32V30; 32V20, 32V05text