Normal Forms of Real Hypersurfaces with Nondegenerate Levi Form

We present a proof of the existence and uniqueness theorem of a normalizing biholomorphic mapping to Chern-Moser normal form. The explicit form of the equation of a chain on a real hyperquadric is obtained. There exists a family of normal forms of real hypersurfaces including Chern-Moser normal form. 0. Introduction Let M be an analytic real hypersurface with nondegenerate Levi form in a complex manifold and p be a point on M . Then it is known that there is a local coordinate system z, z, · · · , z, z ≡ w = u + iv with center at p, where M is locally defined by the equation v = 〈z, z〉+ ∑ s,t≥2 Fst(z, z̄, u), (0.1) where (1) 〈z, z〉 ≡ zz1 + · · · + zze − zze+1 − · · · − zzn for a positive integer e in n 2 ≤ e ≤ n, (2) Fst(z, z̄, u) is a real-analytic function of z, u for each pair (s, t) ∈ N , which satisfies Fst(μz, νz̄, u) = μ νFst(z, z̄, u), for all complex numbers μ, ν, (3) the functions F22, F23, F33 satisfy the following conditions: ∆F22 = ∆ F23 = ∆ F33 = 0, where ∆ ≡ D1D1 + · · ·+DeDe −De+1De+1 − · · · −DnDn, Dk = ∂ ∂zk , Dk = ∂ ∂zk , k = 1, · · · , n. The local coordinate system (0.1) is called normal coordinate. The existence of a normal coordinate is a natural consequence of the following existence theorem of a normalizing biholomorphic mapping to Chern-Moser normal form. E-MAIL: [email protected] MATHEMATICS SUBJECT CLASSIFICATION (1991): PRIMARY:32H99