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Search: All articles in the CMB digital archive with keyword holomorphic

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1. CMB Online first

Gupta, Purvi
A real-analytic nonpolynomially convex isotropic torus with no attached discs
We show by means of an example in $\mathbb C^3$ that Gromov's theorem on the presence of attached holomorphic discs for compact Lagrangian manifolds is not true in the subcritical real-analytic case, even in the absence of an obvious obstruction, i.e, polynomial convexity.

Keywords:polynomial hull, isotropic submanifold, holomorphic disc
Categories:32V40, 32E20, 53D12

2. CMB Online first

Gilligan, Bruce
Levi's problem for pseudoconvex homogeneous manifolds
Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi:G/H \to G/J$ is the holomorphic reduction of $G/H$, i.e., $G/J$ is holomorphically separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$. In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non--constant holomorphic functions.

Keywords:complex homogeneous manifold, plurisubharmonic exhaustion function, holomorphic reduction, Stein manifold, Remmert reduction, Hirschowitz annihilator
Categories:32M10, 32U10, 32A10, 32Q28

3. CMB 2016 (vol 59 pp. 346)

Krantz, Steven
On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains
We study and generalize a classical theorem of L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group.

Keywords:Bers's theorem, algebras of holomorphic functions, noncompact automorphism group, biholomorphic equivalence
Categories:32A38, 30H50, 32A10, 32M99

4. CMB 2015 (vol 59 pp. 303)

Gauthier, P. M.
Non-extendable Zero Sets of Harmonic and Holomorphic Functions
In this paper we study the zero sets of harmonic functions on open sets in $\mathbb{R}^N$ and holomorphic functions on open sets in $\mathbb{C}^N.$ We show that the non-extendability of such zero sets is a generic phenomenon.

Keywords:boundary behaviour, harmonic, holomorphic, zero sets

5. CMB 2015 (vol 58 pp. 350)

Merino-Cruz, Héctor; Wawrzynczyk, Antoni
On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions
We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of functions which are holomorphic on the unit disc $\mathbb{D}$, continuous up to the boundary and of the class $C^{(n)}$ at all points where the function $G$ does not vanish. Here, $G$ refers to a function of the disc algebra without zeros on $\mathbb{D}$. Then we proved that all closed ideals in $\mathfrak{A}_G^n$ with at most countable hull are standard. In the present paper, on the assumption that $G$ is an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the closed ideals $I$ with hull containing $h(G)$ are standard.

Keywords:Banach algebra, disc algebra, holomorphic spaces, standard ideal
Categories:46J15, 46J20, 30H50

6. CMB 2015 (vol 58 pp. 381)

Tang, Xiaomin; Liu, Taishun
The Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1, p_2}$ in $\mathbb{C}^n$
Let $B_{p_1, p_2}=\{z\in\mathbb{C}^n: |z_1|^{p_1}+|z_2|^{p_2}+\cdots+|z_n|^{p_2}\lt 1\}$ be an egg domain in $\mathbb{C}^n$. In this paper, we first characterize the Kobayashi metric on $B_{p_1, p_2}\,(p_1\geq 1, p_2\geq 1)$, and then establish a new type of the classical boundary Schwarz lemma at $z_0\in\partial{B_{p_1, p_2}}$ for holomorphic self-mappings of $B_{p_1, p_2}(p_1\geq 1, p_2\gt 1)$, where $z_0=(e^{i\theta}, 0, \dots, 0)'$ and $\theta\in \mathbb{R}$.

Keywords:holomorphic mapping, Schwarz lemma, Kobayashi metric, egg domain
Categories:32H02, 30C80, 32A30

7. CMB 2011 (vol 56 pp. 31)

Ayuso, Fortuny P.
Derivations and Valuation Rings
A complete characterization of valuation rings closed for a holomorphic derivation is given, following an idea of Seidenberg, in dimension $2$.

Keywords:singular holomorphic foliation, derivation, valuation, valuation ring
Categories:32S65, 13F30, 13A18

8. CMB 2009 (vol 52 pp. 84)

Gauthier, P. M.; Zeron, E. S.
Hartogs' Theorem on Separate Holomorphicity for Projective Spaces
If a mapping of several complex variables into projective space is holomorphic in each pair of variables, then it is globally holomorphic.

Keywords:separately holomorphic, projective space
Categories:32A10, 32D99, 32H99

9. CMB 2008 (vol 51 pp. 618)

Valmorin, V.
Vanishing Theorems in Colombeau Algebras of Generalized Functions
Using a canonical linear embedding of the algebra ${\mathcal G}^{\infty}(\Omega)$ of Colombeau generalized functions in the space of $\overline{\C}$-valued $\C$-linear maps on the space ${\mathcal D}(\Omega)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${\mathcal G}^\infty$. These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

Keywords:Colombeau generalized functions, linear integral operators, generalized holomorphic functions
Categories:32A60, 45P05, 46F30

10. CMB 2007 (vol 50 pp. 579)

Kot, Piotr
$p$-Radial Exceptional Sets and Conformal Mappings
For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary of the unit disc $\partial\mathbb D$ we construct a holomorphic function $f\in\mathbb O(\mathbb D)$ such that \[ \int_{\mathbb D\setminus[0,1]E}|ft|^{p}\,d\mathfrak{L}^{2}<\infty\] and\[ E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}|f(tz)|^{p}\,dt=\infty\Bigr\} .\] In particular if a set $E$ has a measure equal to zero, then a function $f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$.

Keywords:boundary behaviour of holomorphic functions, exceptional sets
Categories:30B30, 30E25

11. CMB 2005 (vol 48 pp. 580)

Kot, Piotr
Exceptional Sets in Hartogs Domains
Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$, defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:|z|<\mu(w),w\in H\}$, where $H$ is an open set in $\mathbb{C}^{n}$ and $\mu$ is a continuous function with positive values in $H$ such that $-\ln\mu$ is a strongly plurisubharmonic function in $H$. Let $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$. For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function $f\in\mathbb{O}(\Omega)$ such that \[ E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}|f(\cdot\,,w)|^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}. \]

Keywords:boundary behaviour of holomorphic functions,, exceptional sets

12. CMB 2005 (vol 48 pp. 500)

Baracco, Luca
Extension of Holomorphic Functions From One Side of a Hypersurface
We give a new proof of former results by G. Zampieri and the author on extension of holomorphic functions from one side $\Omega$ of a real hypersurface $M$ of $\mathbb{C}^n$ in the presence of an analytic disc tangent to $M$, attached to $\bar\Omega$ but not to $M$. Our method enables us to weaken the regularity assumptions both for the hypersurface and the disc.

Keywords:analytic discs, Poisson integral, holomorphic extension
Categories:32D10, 32V25

13. CMB 2001 (vol 44 pp. 126)

Zeron, E. Santillan
Each Copy of the Real Line in $\C^2$ is Removable
Around 1995, Professors Lupacciolu, Chirka and Stout showed that a closed subset of $\C^N$ ($N\geq 2$) is removable for holomorphic functions, if its topological dimension is less than or equal to $N-2$. Besides, they asked whether closed subsets of $\C^2$ homeomorphic to the real line (the simplest 1-dimensional sets) are removable for holomorphic functions. In this paper we propose a positive answer to that question.

Keywords:holomorphic function, removable set

14. CMB 2000 (vol 43 pp. 294)

Bracci, Filippo
Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity
We identify a class of domains of $\C^n$ in which any two commuting holomorphic self-maps have a common fixed point.

Keywords:Holomorphic self-maps, commuting functions, fixed points, Wolff point, Julia's Lemma
Categories:32A10, 32A40, 32H15, 32A30

15. CMB 1999 (vol 42 pp. 139)

Bonet, José; Domański, Paweł; Lindström, Mikael
Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions
Every weakly compact composition operator between weighted Banach spaces $H_v^{\infty}$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_v^{\infty}$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

Keywords:weighted Banach spaces of holomorphic functions, composition operator, compact operator, weakly compact operator
Categories:47B38, 30D55, 46E15

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