Expand all Collapse all | Results 1 - 25 of 64 |
1. CMB Online first
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 |
2. CMB 2014 (vol 57 pp. 697)
On the Monodromy of Milnor Fibers of Hyperplane Arrangements We describe a general setting where the monodromy action on the first
cohomology group of the Milnor fiber of a hyperplane arrangement is
the identity.
Keywords:hyperplane arrangements, Milnor fiber, monodromy, local systems Categories:32S22, 32S55, 32S25, 32S40 |
3. CMB 2014 (vol 57 pp. 658)
Admissibility of Local Systems for some Classes of Line Arrangements Let $\mathcal{A}$ be a line arrangement in the complex
projective plane $\mathbb{P}^2$ and let $M$ be its complement. A rank one
local system $\mathcal{L}$ on $M$ is admissible if roughly speaking
the cohomology groups
$H^m(M,\mathcal{L})$ can be computed directly from the cohomology
algebra $H^{*}(M,\mathbb{C})$. In this work, we give a sufficient
condition for the admissibility of all rank one local systems on
$M$. As a result, we obtain some properties of the characteristic
variety $\mathcal{V}_1(M)$ and the Resonance variety $\mathcal{R}_1(M)$.
Keywords:admissible local system, line arrangement, characteristic variety, multinet, resonance variety Categories:14F99, 32S22, 52C35, 05A18, 05C40, 14H50 |
4. CMB 2014 (vol 57 pp. 673)
Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two If $X$ is a connected complex manifold with $d_X = 2$ that admits a (connected) Lie group $G$
acting transitively as a group of holomorphic transformations, then the action extends to an action of the
complexification $\widehat{G}$ of $G$ on $X$ except when
either the unit disk in the complex plane
or a strictly pseudoconcave homogeneous complex manifold is
the base or fiber of some homogeneous fibration of $X$.
Keywords:homogeneous complex manifold, non-compact dimension two, complexification Category:32M10 |
5. CMB 2013 (vol 57 pp. 870)
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 SeppÃ¤lÃ¤. 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 |
6. CMB 2013 (vol 57 pp. 794)
New Characterizations of the Weighted Composition Operators Between Bloch Type Spaces in the Polydisk |
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 HyvÃ¤rinen and
LindstrÃ¶m in 2012.
Keywords:weighted composition operator, compactness, Bloch type spaces, polydisk, several complex variables Categories:47B38, 47B33, 32A37, 45P05, 47G10 |
7. CMB 2012 (vol 57 pp. 12)
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 |
8. CMB 2011 (vol 56 pp. 593)
On the $p$-norm of an Integral Operator in the Half Plane We give a partial answer to a conjecture of DostaniÄ 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 |
9. CMB 2011 (vol 56 pp. 31)
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 |
10. CMB 2011 (vol 56 pp. 44)
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 |
11. CMB 2011 (vol 55 pp. 108)
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 |
12. CMB 2011 (vol 55 pp. 329)
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 |
13. CMB 2011 (vol 55 pp. 249)
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 |
14. CMB 2011 (vol 55 pp. 242)
Convergence in Capacity In this note we study the convergence of sequences of Monge-AmpÃ¨re measures $\{(dd^cu_s)^n\}$,
where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.
Keywords:complex Monge-AmpÃ¨re operator, convergence in capacity, plurisubharmonic function Categories:32U20, 31C15 |
15. CMB 2011 (vol 55 pp. 441)
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 |
16. CMB 2011 (vol 55 pp. 146)
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 $0
n+1+\alpha$ is particularly interesting. Keywords:Bergman spaces, unit ball, volume measure Category:32A36 |
17. CMB 2011 (vol 54 pp. 230)
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 |
18. CMB 2011 (vol 54 pp. 338)
SzegÃ¶'s Theorem and Uniform Algebras We study SzegÃ¶'s theorem for a uniform algebra.
In particular, we do it for the disc algebra or the bidisc algebra.
Keywords:SzegÃ¶'s theorem, uniform algebras, disc algebra, weighted Bergman space Categories:32A35, 46J15, 60G25 |
19. CMB 2010 (vol 54 pp. 370)
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 |
20. CMB 2010 (vol 54 pp. 56)
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 |
21. CMB 2010 (vol 53 pp. 311)
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 |
22. CMB 2009 (vol 53 pp. 11)
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 |
23. CMB 2009 (vol 53 pp. 23)
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 |
24. CMB 2009 (vol 52 pp. 613)
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 |
25. CMB 2009 (vol 52 pp. 285)
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 |