Lie brackets and singular type of real hypersurfaces

Abstract

This note is the sequel of Barraud and Mazzilli Math Z 248:757-772, 2004, where the regular type of a real hypersurface H in C n (eventually endowed with an almost complex structure) was characterized in terms of Lie brackets of complex tangent vector fields on H . This note extends these results to the singular type.Let J be a smooth almost complex structure on C n , H a smooth real hypersurface, p ∈ H , and let ρ : C n → R be a function such that 0 is a regular value of ρ and H = {ρ = 0}.Let D be the unit disc in C, where the variable will be denoted by t = x + iy. A complex curve will be a J -holomorphic map u : D → C n . The valuation of u at 0 is the integer ν(u) such that u (k) (0) = 0 for k < ν(u) and u (ν(u)) (0) = 0. In the same way, suppose u(0) = p and define ν(ρ(u)) as the vanishing order of ρ(u) at 0.Following [3], letand define the 1-type of H at 0 asThis notion of 1-type is well known in complex analysis and plays a crucial role, for instance, when trying to estimate the regularity of the Cauchy-Riemann equation. D'Angelo proved in [3] that points of finite type form an open set in H . More precisely, he proved that 1 (H, p) ≤ 2 n−1 1 (H, p 0 ) for p close to p 0 . It would be nice to know if this inequality still holds in the almost complex case, or if the finite type condition remains open. This, however, cannot immediately be