Regular type of real hyper-surfaces in (almost) complex manifolds

Abstract

The regular type of a real hyper-surface M in an (almost) complex manifold at some point p is the maximal contact order at p of M with germs of non singular (pseudo) holomorphic disks. The main purpose of this paper is to give two intrinsic characterizations the type : one in terms of Lie brackets of a complex tangent vector field on M, the other in terms of some kind of derivatives of the Levi form. Mathematics Subject Classification (2000): 32T25,32Q60The purpose of this paper is to study order of contact between holomorphic curves (as well pseudoholomorphic curves) and a real hyper-surface. In the integrable case, this invariant is connected to the boundary behavior of the Cauchy-Riemann equations, the Bergman kernel, invariant metrics, etc.. see for example [5,3,4,10,9]. One of the difficulty to use this number in function theory on a domain D, is due to the fact that in general, it was not known how to calculate it with intrinsic "complex geometry" of the boundary of D. In C 2 the situation is clear (see [9]) ; for C n (n > 2), we only know how to compute the order of contact of complex hyper-surface with the natural Lie algebra of the boundary of D (see [2]). In [2], I. Graham and T. Bloom ask how to characterize the regular "type" in a similar intrinsic way with only one complex vector field; T. Bloom (see [1]) succeeds in the pseudoconvex case in C 3 but unfortunately, the result is not valid without the pseudoconvexity hypothesis.In this article, we consider an hyper-surface (pseudoconvex or not) in C n or in R 2n endowed with an almost complex structure, and characterize its "regular one type", by means of Lie brackets of one "complex tangent vector field" on the hyper-surface (see theorem 1), and by means of derivatives of the "levi form" in

Lie brackets and singular type of real hypersurfaces

“…We first recall some notations from [1]. In particular, given a non-vanishing vector field X ∈ C n , it's ( p, q)-th derivative is the vector field (J X) q X p+1 obtained by differentiating X p times in its own direction and then q times in the J X direction (we use the Euclidean connection on C n to define the derivation along X , but if a different connection ∇ would define a different jet, the valuation proposed in Definition 1.1 would be the same).…”

“…From [1,Corollary 11] we deduce that there is a (regular) germ of vector fieldX ∈ TJH at 0, such that…”

“…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.…”

“…The object of [1] was to characterize this type in terms of Lie brackets or jets of J -tangent vectors on H , as this was done by Bloom and Graham [2].…”

“…The main argument here is to turn singular curves into regular ones by considering their graph: this allows the results of [1] to be applied almost directly to this new situation.…”