On a CR family of compact strongly pseudoconvex CR manifolds

We study the simultaneous filling and embedding problem for a CR family of compact strongly pseudoconvex CR manifolds of dimension at least 5. We also derive, as a consequence, the normality of the Stein fibers of the filled-in Stein space under the constant dimensionality assumption of the first Kohn-Rossi cohomology group of the fiber CR manifolds. Two main ingredients for our approach are the work of Catlin on the solution of the partial derivative-equation with mixed boundary conditions and the work of Siu and Ling on the study of the Grauert direct image theory for a (1,1)-convex-concave family of complex spaces.