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Search: MSC category 32 ( Several complex variables and analytic spaces )

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1. CMB Online first

Parlier, Hugo
A short note on short pants
It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in genus, a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound which slightly improve the best known bound.

Keywords:hyperbolic surfaces, geodesics, pants decompositions
Categories:30F10, 32G15, 53C22

2. CMB Online first

Fang, Zhong-Shan; Zhou, Ze-Hua
New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk
We give some new characterizations for compactness of weighted composition operators $uC_\varphi$ acting on Bloch-type spaces in terms of the power of the components of $\varphi,$ where $\varphi$ is a holomorphic self-map of the polydisk $\mathbb{D}^n,$ thus generalizing the results obtained by Hyvärinen and Lindström in 2012.

Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variables
Categories:47B38, 47B33, 32A37, 45P05, 47G10

3. CMB 2012 (vol 57 pp. 12)

Aribi, Amine; Dragomir, Sorin; El Soufi, Ahmad
On the Continuity of the Eigenvalues of a Sublaplacian
We study the behavior of the eigenvalues of a sublaplacian $\Delta_b$ on a compact strictly pseudoconvex CR manifold $M$, as functions on the set ${\mathcal P}_+$ of positively oriented contact forms on $M$ by endowing ${\mathcal P}_+$ with a natural metric topology.

Keywords:CR manifold, contact form, sublaplacian, Fefferman metric
Categories:32V20, 53C56

4. CMB 2011 (vol 56 pp. 593)

Liu, Congwen; Zhou, Lifang
On the $p$-norm of an Integral Operator in the Half Plane
We give a partial answer to a conjecture of Dostanić on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane.

Keywords:Bergman projection, integral operator, $L^p$-norm, the upper half plane
Categories:47B38, 47G10, 32A36

5. 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

6. CMB 2011 (vol 56 pp. 44)

Biswas, Indranil; Dey, Arijit
Polystable Parabolic Principal $G$-Bundles and Hermitian-Einstein Connections
We show that there is a bijective correspondence between the polystable parabolic principal $G$-bundles and solutions of the Hermitian-Einstein equation.

Keywords:ramified principal bundle, parabolic principal bundle, Hitchin-Kobayashi correspondence, polystability
Categories:32L04, 53C07

7. CMB 2011 (vol 55 pp. 108)

Guler, Dincer
On Segre Forms of Positive Vector Bundles
The goal of this note is to prove that the signed Segre forms of Griffiths' positive vector bundles are positive.

Categories:53C55, 32L05

8. CMB 2011 (vol 55 pp. 329)

Kamiya, Shigeyasu; Parker, John R.; Thompson, James M.
Non-Discrete Complex Hyperbolic Triangle Groups of Type $(n,n, \infty;k)$
A complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type $(n,n,\infty;k)$.

Keywords:complex hyperbolic triangle group
Categories:51M10, 32M15, 53C55, 53C35

9. CMB 2011 (vol 55 pp. 249)

Chang, Der-Chen; Li, Bao Qin
Description of Entire Solutions of Eiconal Type Equations
The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases, where $X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$, $X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$ are linearly independent operators with $p_j$ being arbitrary polynomials in $\mathbf{C}^2$.

Keywords:entire solution, eiconal equation, polynomial, transcendental function
Categories:32A15, 35F20

10. CMB 2011 (vol 55 pp. 242)

Cegrell, Urban
Convergence in Capacity
In this note we study the convergence of sequences of Monge-Ampère measures $\{(dd^cu_s)^n\}$, where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.

Keywords:complex Monge-Ampère operator, convergence in capacity, plurisubharmonic function
Categories:32U20, 31C15

11. CMB 2011 (vol 55 pp. 441)

Zorboska, Nina
Univalently Induced, Closed Range, Composition Operators on the Bloch-type Spaces
While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces $B^{\alpha}$ with $\alpha \ne 1$ are the ones induced by a disc automorphism.

Keywords:composition operators, Bloch-type spaces, closed range, univalent
Categories:47B35, 32A18

12. CMB 2011 (vol 55 pp. 146)

Li, Songxiao; Wulan, Hasi; Zhu, Kehe
A Characterization of Bergman Spaces on the Unit Ball of ${\mathbb C}^n$. II
It has been shown that a holomorphic function $f$ in the unit ball $\mathbb{B}_n$ of ${\mathbb C}_n$ belongs to the weighted Bergman space $A^p_\alpha$, $p>n+1+\alpha$, if and only if the function $|f(z)-f(w)|/|1-\langle z,w\rangle|$ is in $L^p(\mathbb{B}_n\times\mathbb{B}_n,dv_\beta \times dv_\beta)$, where $\beta=(p+\alpha-n-1)/2$ and $dv_\beta(z)= (1-|z|^2)^\beta\,dv(z)$. In this paper we consider the range $0n+1+\alpha$ is particularly interesting.

Keywords:Bergman spaces, unit ball, volume measure
Category:32A36

13. CMB 2011 (vol 54 pp. 230)

Clouâtre, Raphaël
Universal Power Series in $\mathbb{C}^N$
We establish the existence of power series in $\mathbb{C}^N$ with the property that the subsequences of the sequence of partial sums uniformly approach any holomorphic function on any well chosen compact subset outside the set of convergence of the series. We also show that, in a certain sense, most series enjoy this property.

Categories:32A05, 32E30

14. CMB 2011 (vol 54 pp. 338)

Nakazi, Takahiko
Szegö's Theorem and Uniform Algebras
We study Szegö's theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra.

Keywords:Szegö's theorem, uniform algebras, disc algebra, weighted Bergman space
Categories:32A35, 46J15, 60G25

15. CMB 2010 (vol 54 pp. 370)

Stout, Edgar Lee
Manifold-Valued Holomorphic Approximation
This note considers the problem of approximating continuous maps from sets in complex spaces into complex manifolds by holomorphic maps.

Category:32E20

16. CMB 2010 (vol 54 pp. 56)

Dinh, Thi Anh Thu
Characteristic Varieties for a Class of Line Arrangements
Let $\mathcal{A}$ be a line arrangement in the complex projective plane $\mathbb{P}^2$, having the points of multiplicity $\geq 3$ situated on two lines in $\mathcal{A}$, say $H_0$ and $H_{\infty}$. Then we show that the non-local irreducible components of the first resonance variety $\mathcal{R}_1(\mathcal{A})$ are 2-dimensional and correspond to parallelograms $\mathcal{P}$ in $\mathbb{C}^2=\mathbb{P}^2 \setminus H_{\infty}$ whose sides are in $\mathcal{A}$ and for which $H_0$ is a diagonal.

Keywords:local system, line arrangement, characteristic variety, resonance variety
Categories:14C21, 14F99, 32S22, 14E05, 14H50

17. CMB 2010 (vol 53 pp. 311)

Jasiczak, Michał
Remark on Zero Sets of Holomorphic Functions in Convex Domains of Finite Type
We prove that if the $(1,1)$-current of integration on an analytic subvariety $V\subset D$ satisfies the uniform Blaschke condition, then $V$ is the zero set of a holomorphic function $f$ such that $\log |f|$ is a function of bounded mean oscillation in $bD$. The domain $D$ is assumed to be smoothly bounded and of finite d'Angelo type. The proof amounts to non-isotropic estimates for a solution to the $\overline{\partial}$-equation for Carleson measures.

Categories:32A60, 32A35, 32F18

18. CMB 2009 (vol 53 pp. 11)

Burke, Maxim R.
Approximation and Interpolation by Entire Functions of Several Variables
Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon \mathbb R^n\to\mathbb R$ be positive and continuous. For any unbounded nondecreasing sequence $\{c_k\}$ of nonnegative real numbers and for any sequence without accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that \begin{align*} &|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k, k=0,1,2,\dots, \\ &g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k, m,k=0,1,2,\dots. \end{align*} This is a version for functions of several variables of the case $n=1$ due to L. Hoischen.

Keywords:entire function, complex approximation, interpolation, several complex variables
Category:32A15

19. CMB 2009 (vol 53 pp. 23)

Chen, Huaihui; Zhang, Minzhu
Boundedness From Below of Multiplication Operators Between $\alpha$-Bloch Spaces
In this paper, the boundedness from below of multiplication operators between $\alpha$-Bloch spaces $\mathcal B^\alpha$, $\alpha\gt 0$, on the unit disk $D$ is studied completely. For a bounded multiplication operator $M_u\colon \mathcal B^\alpha\to\mathcal B^\beta$, defined by $M_uf=uf$ for $f\in\mathcal B^\alpha$, we prove the following result: (i) If $0\lt \beta\lt \alpha$, or $0\lt \alpha\le1$ and $\alpha\lt \beta$, $M_u$ is not bounded below; (ii) if $0\lt \alpha=\beta\le1$, $M_u$ is bounded below if and only if $\liminf_{z\to\partial D}|u(z)|\gt 0$; (iii) if $1\lt \alpha\le\beta$, $M_u$ is bounded below if and only if there exist a $\delta\gt 0$ and a positive $r\lt 1$ such that for every point $z\in D$ there is a point $z'\in D$ with the property $d(z',z)\lt r$ and $(1-|z'|^2)^{\beta-\alpha}|u(z')|\ge\delta$, where $d(\cdot,\cdot)$ denotes the pseudo-distance on $D$.

Keywords:$\alpha$-Bloch function, multiplication operator
Categories:32A18, 30H05

20. CMB 2009 (vol 52 pp. 613)

Wulan, Hasi; Zhu, Kehe
Lipschitz Type Characterizations for Bergman Spaces
We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.

Keywords:Bergman spaces, hyperbolic metric, Lipschitz condition
Category:32A36

21. CMB 2009 (vol 52 pp. 285)

Parker, John R.; Platis, Ioannis D.
Global Geometrical Coordinates on Falbel's Cross-Ratio Variety
Falbel has shown that four pairwise distinct points on the boundary of a complex hyperbolic $2$-space are completely determined, up to conjugation in ${\rm PU}(2,1)$, by three complex cross-ratios satisfying two real equations. We give global geometrical coordinates on the resulting variety.

Categories:32G05, 32M05

22. CMB 2009 (vol 52 pp. 175)

Biswas, Indranil
Connections on a Parabolic Principal Bundle, II
In \emph{Connections on a parabolic principal bundle over a curve, I} we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.

Keywords:Parabolic bundle, Atiyah exact sequence, connection
Categories:32L05, 14F05

23. CMB 2009 (vol 52 pp. 154)

Ye, Yasheng; Ru, Min
A Big Picard Theorem for Holomorphic Maps into Complex Projective Space
We prove a big Picard type extension theorem for holomorphic maps $f\from X-A \rightarrow M$, where $X$ is a complex manifold, $A$ is an analytic subvariety of $X$, and $M$ is the complement of the union of a set of hyperplanes in ${\Bbb P}^n$ but is not necessarily hyperbolically imbedded in ${\Bbb P}^n$.

Category:32H30

24. 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

25. CMB 2008 (vol 51 pp. 481)

Bayart, Frédéric
Universal Inner Functions on the Ball
It is shown that given any sequence of automorphisms $(\phi_k)_k$ of the unit ball $\bn$ of $\cn$ such that $\|\phi_k(0)\|$ tends to $1$, there exists an inner function $I$ such that the family of ``non-Euclidean translates" $(I\circ\phi_k)_k$ is locally uniformly dense in the unit ball of $H^\infty(\bn)$.

Keywords:inner functions, automorphisms of the ball, universality
Categories:32A35, 30D50, 47B38
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