Hartogs-type extension for tube-like domains in $$\mathbb C^2$$ C 2

Abstract

In this paper we consider the Hartogs-type extension problem for unbounded domains in C 2 . An easy necessary condition for a domain to be of Hartogs-type is that there is no a closed (in C 2 ) complex variety of codimension one in the domain which is given by a holomorphic function smooth up to the boundary. The question is, how far this necessary condition is from the sufficient one? To show how complicated this question is, we give a class of tube-like domains which contain a complex line in the boundary which are either of Hartogs-type or not, depending on how the complex line is positioned with respect to the domain.

On the Hartogs extension theorem for unbounded domains in $\mathbb{C}^n$

“…Remark 5.3 a) In some cases (for example if M is strictly pseudoconvex at every point and Ω is the pseudoconvex side) the CR distributions we find to be obstructions to Hartogs extension are smooth on M . In general this cannot always be achieved because of examples constructed in [5], see Corollary 1. Most of the proof of Theorem 4.1 is easily generalised to Stein manifold.…”

“…The classical Hartogs extension theorem made an important influence not only on Complex Analysis, but also on other areas of mathematics, like Algebraic Geometry or Partial Differential Equations. The theorem still inspires researchers and there is a renewed interest in recent years: Harz-Shcherbina-Tomassini [15,16], Øvrelid-Vassiliadou [28], Damiano-Struppa-A.Vajiac-M.Vajiac [9], Palamodov [29], Ohsawa [27], Coltoiu-Ruppenthal [8], Lewandowski [21], and papers by the authors with other colleagues [3,4,5,6], [24,25].…”

“…Obviously extension still fails if Ω contains a complex hypersurface. The case where Ω contains a complex hypersurface but Ω does not is more delicate: Examples constructed in [5] by the first two authors and Z. S lodkowski show that simultaneous extension of smooth CR functions to Ω may be valid or not, depending on a finer geometry of intersection of the complex hypersurface and the boundary.…”

“…For more results and questions on particular domains (bounded or unbounded) we refer to [3,4,5,6,11,15,16,22,33]. It may also be interesting to study domains obtained by intersecting smoothly bounded domains in CP 2 with C 2 , for example with respect to extension from parts of the boundary, see [26] for domains in C 2 and further references.…”

$${\bar{\partial }}$$ ∂ ¯ -Problem in Fiber Bundles for Decreasing (0, 1)-Forms

“…The theorems like Theorem 1 or from [7] are very useful in the Hartogs-type extension (see [12,13]) of holomorphic or Cauchy-Riemann functions in a wide class of complex manifolds, see [2][3][4][5][6]9,10,15,16,18]. Applications are given in [7] and we do not repeat them here.…”