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On the Kuranishi family of deformations of complex structures over a tubular neighborhood of a strongly pseudo convex boundary with complex dimension 3.

It was Kuranishi who initiated the deformation theory of isolated singularities from the point of view of CR-structures (see [12]). This approach was developed by several authors ([l]-[6], [13], [14]). Especially, inspired by Kuranishi's approach, the versal family of strongly pseudo convex domains with complex dimension ^4 was constructed by introducing the so called new type Neumann operator ([3]-[4]). And furthermore in [14], a deep relation between the versal family of isolated singularities constructed by Grauert, and the family defined by our versal family of CR-structures, is shown. The motivation of this paper is to construct a versal family of strongly pseudo convex domains with complex dimension 3. Unfortunately, the complex dimension 3 case, in general, is still a problem. However, if we impose several assumptions, it may be possible. In this paper, we discuss this and construct a versal family of tubular neighborhoods of strongly pseudo convex boundaries with complex dimension 3 under an assumption. For this purpose, we explain the Situation more precisely. Let 7V be a complex manifold with complex dimension 3 and let Ω be a strongly pseudo convex domain with smooth boundary b . Let Qad be a tubular neighborhood of bQ. As we know, the deformation problem is reduced to solving the non linear partial differential equation Ρ(φ) = 0 (see [11]). In the 3 dimensional tubular neighborhood case,