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We survey some recent results concerning the behavior of the contact structure defined on the boundary of a complex isolated hypersurface singularity or on the boundary at infinity of a complex polynomial.

Contact geometry and complex surfaces.

Hansjörg Geiges, Jesús Gonzalo (1995)

Inventiones mathematicae

Contact geometry and CR-structures on spheres

John Bland, Tom Duchamp (1995)

Banach Center Publications

A normal form for small CR-deformations of the standard CR-structure on the (2n+1)-sphere is presented. The space of normal forms is parameterized by a single function on the sphere. For n>1, the normal form is used to obtain explicit embeddings into $ℂ^{n + 1}$ . For n=1, the cohomological obstruction to embeddability is identified.

Continuation of holomorphic functions with growth conditions and some of its applications

Alexander V. Abanin, Pham Trong Tien (2010)

Studia Mathematica

We prove a generalization of the well-known Hörmander theorem on continuation of holomorphic functions with growth conditions from complex planes in $ℂ^{p}$ into the whole $ℂ^{p}$ . We apply this result to construct special families of entire functions playing an important role in convolution equations, interpolation and extension of infinitely differentiable functions from closed sets. These families, in their turn, are used to study optimal or canonical, in a certain sense, weight sequences defining inductive...

Continuation of holomorphic solutions to convolution equations in complex domains

Ryuichi Ishimura, Jun-ichi Okada, Yasunori Okada (2000)

Annales Polonici Mathematici

For an analytic functional $S$ on $ℂ^{n}$ , we study the homogeneous convolution equation S * f = 0 with the holomorphic function f defined on an open set in $ℂ^{n}$ . We determine the directions in which every solution can be continued analytically, by using the characteristic set.

Continuation of positive, closed currents and analytic sets.

Manfred Rapp (1993)

Mathematische Zeitschrift

Continuing 1 -dimensional Analytic Sets.

H. ALEXANDER (1971)

Mathematische Annalen

Continuing Analytic Sets Across IRn.

Joe Becker (1971)

Mathematische Annalen

Continuity and convergence properties of extremal interpolating disks.

Pascal J. Thomas (1995)

Publicacions Matemàtiques

Let a be a sequence of points in the unit ball of Cn. Eric Amar and the author have introduced the nonnegative quantity ρ(a) = infα infk Πj:j≠k dG(αj, αk), where dG is the Gleason distance in the unit disk and the first infimum is taken over all sequences α in the unit disk which map to a by a map from the disk to the ball.The value of ρ(a) is related to whether a is an interpolating sequence with respect to analytic disks passing through it, and if a is an interpolating sequence in the ball, then...

Continuity of intersection of analytic sets

P. Tworzewski, T. Winiarski (1983)

Annales Polonici Mathematici

Continuity of plurisubharmonic envelopes

Nihat Gokhan Gogus (2005)

Annales Polonici Mathematici

Let D be a domain in ℂⁿ. The plurisubharmonic envelope of a function φ ∈ C(D̅) is the supremum of all plurisubharmonic functions which are not greater than φ on D. A bounded domain D is called c-regular if the envelope of every function φ ∈ C(D̅) is continuous on D and extends continuously to D̅. The purpose of this paper is to give a complete characterization of c-regular domains in terms of Jensen measures.

Continuity of the relative extremal function on analytic varieties in ℂⁿ

Frank Wikström (2005)

Annales Polonici Mathematici

Let V be an analytic variety in a domain Ω ⊂ ℂⁿ and let K ⊂ ⊂ V be a closed subset. By studying Jensen measures for certain classes of plurisubharmonic functions on V, we prove that the relative extremal function $ω_{K}$ is continuous on V if Ω is hyperconvex and K is regular.

Continuity of total number of intersection

T. Winiarski (1986)

Annales Polonici Mathematici

Continuous and $L^{p}$ estimates for the complex Monge-Ampère equation on bounded domains in $ℂ^{n}$ .

Darko, Patrick W. (2002)

International Journal of Mathematics and Mathematical Sciences

Continuous division of differential operators

Herwig Hauser, Luis Narváez-Macarro (2001)

Annales de l’institut Fourier

In this paper, we give a new proof of the continuity of division by differential operators with analytic coefficients, originally proved by Mebkhout and the second author. Our methods come from the proof of the Constant Rank Theorem for analytic maps between power series spaces, given by Müller and the first author.

Continuous linear extension operators on spaces of holomorphic functions on closed subgroups of a complex Lie group

Do Duc Thai, Dinh Huy Hoang (1999)

Annales Polonici Mathematici

We show that the restriction operator of the space of holomorphic functions on a complex Lie group to an analytic subset V has a continuous linear right inverse if it is surjective and if V is a finite branched cover over a connected closed subgroup Γ of G. Moreover, we show that if Γ and G are complex Lie groups and V ⊂ Γ × G is an analytic set such that the canonical projection $π_{1} : V \to Γ$ is finite and proper, then $R_{V} : O (Γ \times G) \to I m R_{V} \subset O (V)$ has a right inverse

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