Complex Structures and Characterization of the Space

We give a characterization of the complex Euclidean space C n , proving a global Newlander-Nirenberg's theorem on the representability of complex structures close to the standard complex structure on C n. Let M be a smooth real manifold of dimension 2n. An almost complex structure J on M is a tensor field of type (1,1) on M (that is a section of End(T M)) satisfying J 2 = −I. It is called integrable if for any point p ∈ M there exists an open neighborhood U of p and a diffeomorphism z : U −→ B between U and the unit ball of C n such that (z *)(J) := dz • J • dz −1 = J st where J st denote the standard complex structure of C n. In other words the coordinate z is biholomorphic with respect to J and J st and M admits local complex holomorphic coordinates near every point. A structure J is called formally integrable if N vanishes at every point of M. The fundamental theorem of Newlander-Nirenberg [8] states that the formal integrability is equivalent to integra-bility. In the present note we prove a global version of this result for almost complex structures defined on the whole space R 2n. Our approach is a modification of the classical Hörmander's proof [6] of the Newlander-Nirenberg theorem. Let J be an integrable almost complex structure of class C 3 on R 2n. We assume that J − J st C 3 (R 2n) ≤ λ (1.1) where λ > 0 is small enough. Fix a basis ω j , j = 1,. .. , n of differential forms of type (1,0) with respect to J on R 2n. Since J is close to J st we can choose ω j in the form ω j = dz j + n k=1 b jk dz k where the C 3 coefficients b jk are small enough. Consider the Hermitian metric ds 2 = δ ij ω i ⊗ω j on R 2n (here δ ij denotes the Kroneker symbol). This metric is compatible with J. If the (1,0) vector fields ω j * = ∂/∂z j + k a jk ∂/∂z k are dual to the forms ω j , then they form an orthonormal basis of the tangent space T (1,0) M of (1, 0) vector fields with respect to ds 2. Since …