RIGID SPHERICAL HYPERSURFACES IN C 2 Abdelaziz

In this paper we describe explicitly one class of real-analytic hypersurfaces in C2 rigid and spherical at the origin. 206 Abdelaziz Chaouech


Introduction
A real-analytic hypersurface M in C 2 is called rigid if it is given by an equation of the form r(w, w, z, z) =: Imw + F (z, z) = 0, where F is a real-analytic function such that: F (0, 0) = ∂F ∂z (0, 0) = 0.
In this paper we study the real-analytic hypersurfaces M in C 2 rigid and spherical at the origin, i.e. there exists a local biholomorphic which maps M to the euclidean unit sphere.We note that recently A. Isaev [4] has given a characterization of spherical rigid real hypersurfaces in C n (n ≥ 2) in terms of a certain system of differential equations for a defining function of such hypersurfaces, but this does not permit to describe the spherical rigid real hypersurfaces even if in C 2 .Nowadays these hypersurfaces are not known.The only examples have been given by N. Stanton [5] (see also [6]).More recently, B. Coupet and A. Sukhov [3] have described the spherical hypersurfaces of the form: Imw + P (z, z) = 0, where P is a non-identically zero subharmonic homogeneous polynomial without purely harmonic terms.
The goal of this paper is to give one description of one class of real-analytic hypersurfaces in C 2 rigid and spherical at the origin.

Prelimenaries and results
Let M be a hypersurface in C 2 , strictly pseudoconvex at the origin, defined by: , where ϕ is a real-analytic function.
Without any loss of generality, we may assume that According to a theorem of N. Stanton [5] (see theorem 1.7) there exists an holomorphic change of coordinates ψ of the form (w, g(z)) defined in a neighborhood V of the origin and such that ψ(M ∩ V ) is defined by: where b is a real-analytic function.
Theorem.Let M be a hypersurface in C 2 defined by M =: {Rew + ϕ(z, z) = 0} , where ϕ(z, z) = |z| 2 +|z| 4 b(z, z) and b being a real-analytic function in a neighbourhood of the origin.

Suppose that
∂b ∂z (0, 0) = 0. Then M is spherical at the origin if and only if ϕ is given by one of the functions: Proof.Let F = (F 1 , F 2 ) be a local biholomorphism at the origin which maps M to the euclidean unit sphere: (w, z) ∈ C 2 : ρ(w, z) =: Rew + |z| 2 = 0 .
We may assume that F 1 (0, 0) = F 2 (0, 0) = 0 and By conjugating F with some automorphism of the euclidean unit ball of C 2 , we may also assume that ∂F 2 ∂w (0, 0) = 0.The principal idea of the proof is to determine explicitly F by solving a system of partial differential equations.We note that the real dimension of the lie algebra of holomorphic tangent vector fields on the unit sphere is equal to 8 (see E. Cartan [1]).
We now proceed in three steps.
We are now in order to solve the system of differential equations (7) and (8).
There are four cases to consider.For example, we suppose that F is given by ii), i.e.F (w, z) , where γ ∈ C * and Since the image F (M ) is contained in the unit sphere: (w, z) ∈ C 2 : Rew + |z| 2 = 0 near (0, 0), we have By following the same way we obtain the other cases.
To end the proof of the theorem it remains to show that: ∂b ∂z (0, 0) and Let's return to the indentities( 5) and (6).From the first step we have: ∂ n F 1 ∂z n (0, 0) = 0, ∀n ≥ 0, then, from (5) we deduce that: (0, 0) = 0, ∀n ≥ 1, So, in a neighbourhood fo the origin, we can write: The idea to prove λ = 1 2i ∂b ∂z (0, 0) and α 5 = − β 0 2 is to observe that the terms of degree less or equal to 4 on z, z in the left hand-side of (1) are null.
First, we observe that from ( 5) and ( 19) we have: (21) A 12 = −iλ and a 2 = −iα 4 Next, from ( 6) and (20) we have: We are now in order to collect the terms of degree less or equal to 4 on z, z in the left hand-side of (1).