Completion of Akahori’s construction of the versal family of strongly pseudoconvex CR structures

Abstract:

Let $M$ be a compact smooth boundary of a strongly pseudo-convex domain of a complex manifold $N$ with dim $N \geqslant 4$. We established a sharp a priori estimate for the Laplacian operator associated with Akahori’s subcomplex of the $T’{N_{|M}}$-valued ${\bar \partial _b}$-complex to construct the complex analytic versal family (in the sense of Kuranishi) of $CR$ structures of class ${C^\infty }$ on $M$.