Kuranishi type Moduli Spaces for proper CR submersions fibering over the circle

Kuranishi's fundamental result (1962) associates to any compact complex manifold $X_0$ a finite-dimensional analytic space which has to be thought of as a local moduli space of complex structures close to $X_0$. In this paper, we give an analogous statement for Levi-flat CR manifolds fibering properly over the circle by associating to any such $\mathcal X_0$ the loop space of a finite-dimensional analytic space which serves as a local moduli space of CR structures close to $\mathcal X_0$. We then develop in this context a Kodaira-Spencer deformation theory making clear the likenesses as well as the differences with the classical case. The article ends with applications and examples.