On the tangential holomorphic vector fields vanishing at an infinite type point

Abstract

Let (M, p) be a C ∞ smooth non-Leviflat CR hypersurface germ in C 2 where p is of infinite type. The purpose of this article is to investigate the holomorphic vector fields tangent to (M, p) vanishing at p.

On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type

“…It is shown in [16,22] (see also Corollary 6.4 in this paper) that the second example shows (2) (3) and (1) (4) in the two-dimensional case. The higher dimensional case can be easily shown.…”

“…It is easy to check that the examples of hypersurface constructed in [3,9,16,22] do not admit any N -canonical coordinates. More exactly, we will give equivalence conditions in more restricted cases in Sects.…”

“…It has been shown in [7,11,19] that when M is real analytic, the above two conditions are equivalent (in this case, the curve γ ∞ is contained in M). Moreover, in the case of smooth M, this equivalence is also shown in the formal series sense in [8,16]. But, in general, the sufficient direction is not true.…”

“…But, in general, the sufficient direction is not true. Indeed, the nonexistence of the curve γ ∞ in (1.2) is shown in the case of some real hypersurfaces constructed in [3,9,16,22]. Understanding flatness on hypersurfaces of infinite type has been recognized to be a delicate issue.…”

A note on pseudoconvex hypersurfaces of infinite type in $\mathbb C^n$

“…This question plays a crucial role in the regularity of∂-Neumann problems over pseudoconvex domains (see [D'A82, Cat83,Cat84,Cat87,DK99], and the references therein). The main results around this question are due to T. Bloom and I. Graham [BG77], L. Lempert and J. P. D'Angelo [D'A93, Lem86], the first author and B. Stensønes [FS12], the first author, L. Lee and Y. Zhang [FLZ14], and K.-T. Kim and the second author [KN15].…”