Embedding of Pseudoconvex Cr Manifolds of Levi-forms with One Degenerate Eigenvalue

Suppose that M is an abstract smoothly bounded orientable CR manifold of dimension 2n − 1 with a given integrable CR structure S of dimension n− 1. Since M is orientable, there are a smooth real nonvanishing 1-form η and a smooth real vector field X0 on M so that η(X) = 0 for all X ∈ S and η(X0) = 1. We define the Levi form of S on M by iη([X ′, X ′′]), X ′, X ′′ ∈ S. We may assume that M ⊂ M̃ , in C∞ sense, where M̃ is a smooth manifold. Ever since the discovery of non-realizable CR-structures on M with dimRM = 3 [15], the question of local embeddability of M as a real hypersurface in Cn has been one of the main interests in CR-geometry. In [12], Jacobowitz and Treves also showed that there is (M,S), of nondegenerate Levi-form with only one positive eigenvalue, which can not be realizable. However, Kuranishi [13] showed that (M,S) can be realizable provided M is strongly pseudoconvex and dimRM ≥ 9. Later, Akahori [1], Webster [17] proved the same result when M is strongly pseudoconvex and dimRM ≥ 7. Recently, under certain conditions on the Levi-form, Catlin [5] extended the given CR structure on M to an integrable almost complex structure on a 2n-dimensional manifold Ω with boundary so that the extension is smooth up to the boundary and so M lies in bΩ. This leads to a solution of the local embedding problem provided M is pseudoconvex and the Levi-form of M has at least three positive eigenvalues (so dimRM ≥ 7). In this paper, we consider a local embedding problem of a given CR structure on M when M is a pseudoconvex CR manifold of finite type with one degenerate eigenvalue and dimRM ≥ 7. For given positive continuous