Expand all Collapse all | Results 1 - 25 of 48 |
1. CJM Online first
Pontryagin's maximum principle for the Loewner equation in higher dimensions In this paper we develop a variational method for the Loewner
equation in higher dimensions. As a result we obtain a version of Pontryagin's
maximum principle from optimal control theory for the Loewner equation in
several complex variables. Based on recent work of Arosio, Bracci and
Wold,
we then apply our version of the Pontryagin maximum
principle to obtain first-order necessary conditions for the extremal
mappings for a wide class
of extremal problems over the set of normalized biholomorphic
mappings on the unit ball in $\mathbb{C}^n$.
Keywords:univalent function, Loewner's equation Categories:32H02, 30C55, 49K15 |
2. CJM 2014 (vol 66 pp. 961)
Moduli Spaces of Vector Bundles over a Real Curve: $\mathbb Z/2$-Betti Numbers Moduli spaces of real bundles over a real curve arise naturally
as Lagrangian submanifolds of the moduli space of semi-stable
bundles over a complex curve. In this paper, we adapt the methods
of Atiyah-Bott's ``Yang-Mills over a Riemann Surface'' to compute
$\mathbb Z/2$-Betti numbers of these spaces.
Keywords:cohomology of moduli spaces, holomorphic vector bundles Categories:32L05, 14P25 |
3. CJM Online first
Generalized KÃ¤hler--Einstein Metrics and Energy Functionals In this paper, we consider a generalized
KÃ¤hler-Einstein equation on KÃ¤hler manifold $M$. Using the
twisted $\mathcal K$-energy introduced by Song and Tian, we show
that the existence of generalized KÃ¤hler-Einstein metrics with
semi-positive twisting $(1, 1)$-form $\theta $ is also closely
related to the properness of the twisted $\mathcal K$-energy
functional. Under the condition that the twisting form $\theta $ is
strictly positive at a point or $M$ admits no nontrivial Hamiltonian
holomorphic vector field, we prove that the existence of generalized
KÃ¤hler-Einstein metric implies a Moser-Trudinger type inequality.
Keywords:complex Monge--AmpÃ¨re equation, energy functional, generalized KÃ¤hler--Einstein metric, Moser--Trudinger type inequality Categories:53C55, 32W20 |
4. CJM 2012 (vol 65 pp. 808)
On Hessian Limit Directions along Gradient Trajectories Given a non-oscillating gradient trajectory $|\gamma|$ of a real analytic function $f$,
we show that the limit $\nu$ of the secants at the limit point
$\mathbf{0}$
of $|\gamma|$ along the trajectory
$|\gamma|$ is an eigen-vector of the limit of the direction of the
Hessian matrix $\operatorname{Hess} (f)$ at $\mathbf{0}$
along $|\gamma|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce
some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is
of metric nature and still holds in a general Riemannian analytic setting.
Keywords:gradient trajectories, non-oscillation, limit of Hessian directions, limit of secants, trajectories at infinity Categories:34A26, 34C08, 32Bxx, 32Sxx |
5. CJM 2012 (vol 66 pp. 197)
On Hyperbolicity of Domains with Strictly Pseudoconvex Ends This article establishes a sufficient condition for Kobayashi
hyperbolicity of unbounded domains in terms of curvature.
Specifically, when $\Omega\subset{\mathbb C}^{n}$ corresponds to a
sub-level set of a smooth, real-valued function $\Psi$, such that the
form $\omega = {\bf i}\partial\bar{\partial}\Psi$ is KÃ¤hler and
has bounded curvature outside a bounded subset, then this domain
admits a hermitian metric of strictly negative holomorphic sectional
curvature.
Keywords:Kobayashi-hyperbolicity, KÃ¤hler metric, plurisubharmonic function Categories:32Q45, 32Q35 |
6. CJM 2012 (vol 65 pp. 721)
Tameness of Complex Dimension in a Real Analytic Set Given a real analytic set $X$ in a complex manifold and a positive
integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which
there exists a germ of a complex analytic set of dimension $d$ contained in $X$.
It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$.
Keywords:complex dimension, finite type, semianalytic set, tameness Categories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15 |
7. CJM 2012 (vol 64 pp. 721)
Analysis of the Brylinski-Kostant Model for Spherical Minimal Representations We revisit with another view point the construction by R. Brylinski
and B. Kostant of minimal representations of simple Lie groups. We
start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$
a homogeneous polynomial of degree 4 on $V$.
The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$,
the conformal group of the pair $(V,Q)$, in a finite dimensional
representation space.
By a generalized Kantor-Koecher-Tits construction we obtain a complex
simple Lie algebra $\mathfrak g$, and furthermore a real
form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie
group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak
g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic
functions defined on the manifold $\Xi $.
Keywords:minimal representation, Kantor-Koecher-Tits construction, Jordan algebra, Bernstein identity, Meijer $G$-function Categories:17C36, 22E46, 32M15, 33C80 |
8. CJM 2011 (vol 64 pp. 1329)
Composition Operators Induced by Analytic Maps to the Polydisk We study properties of composition operators
induced by symbols acting from the unit disk to the polydisk.
This result will be involved in the investigation
of weighted composition operators on the Hardy space on the unit disk
and moreover be concerned with composition operators acting
from the Bergman space to the Hardy space on the unit disk.
Keywords:composition operators, Hardy spaces, polydisk Categories:47B33, 32A35, 30H10 |
9. CJM 2011 (vol 64 pp. 429)
Holomorphic Mappings between Domains in $\mathbb C^2$ An extension theorem for holomorphic mappings between two domains in
$\mathbb C^2$ is proved under purely local hypotheses.
Keywords:reflection principle, Segre varieties Categories:32H40, 32H40 |
10. CJM 2011 (vol 63 pp. 755)
On the Geometry of the Moduli Space of Real Binary Octics The moduli space of smooth real binary octics has five connected
components. They parametrize the real binary octics whose defining
equations have $0,\dots,4$ complex-conjugate pairs of roots
respectively. We show that each of these five components has a real
hyperbolic structure in the sense that each is isomorphic as a
real-analytic manifold to the quotient of an open dense subset of
$5$-dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an
arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are
commensurable to discrete hyperbolic reflection groups, and the
Vinberg diagrams of the latter are computed.
Keywords:real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithm Categories:32G13, 32G20, 14D05, 14D20 |
11. CJM 2011 (vol 63 pp. 1038)
Critical Points and Resonance of Hyperplane Arrangements If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement
$\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship
between the critical set of $\Phi_\lambda$, the variety defined by the vanishing
of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$.
For arrangements satisfying certain conditions, we show that if $\lambda$ is
resonant in dimension $p$, then the critical set
of $\Phi_\lambda$ has codimension
at most $p$. These include all free arrangements and all rank $3$ arrangements.
Keywords:hyperplane arrangement, master function, resonant weights, critical set Categories:32S22, 55N25, 52C35 |
12. CJM 2010 (vol 62 pp. 1276)
A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball |
A Generalized Poisson Transform of an $L^{p}$-Function over the Shilov Boundary of the $n$-Dimensional Lie Ball
Let $\mathcal{D}$ be the $n$-dimensional Lie ball and let
$\mathbf{B}(S)$ be the space of hyperfunctions on the Shilov
boundary $S$ of $\mathcal{D}$.
The aim of this paper is to give a
necessary and sufficient condition on the generalized Poisson
transform $P_{l,\lambda}f$ of an element $f$ in the space
$\mathbf{B}(S)$ for $f$ to be in $ L^{p}(S)$, $1 > p > \infty.$
Namely, if $F$ is the Poisson transform of some $f\in
\mathbf{B}(S)$ (i.e., $F=P_{l,\lambda}f$), then for any
$l\in \mathbb{Z}$ and $\lambda\in \mathbb{C}$ such that
$\mathcal{R}e[i\lambda] > \frac{n}{2}-1$, we show that $f\in L^{p}(S)$ if and
only if $f$ satisfies the growth condition
$$
\|F\|_{\lambda,p}=\sup_{0\leq r
< 1}(1-r^{2})^{\mathcal{R}e[i\lambda]-\frac{n}{2}+l}\Big[\int_{S}|F(ru)|^{p}du
\Big]^{\frac{1}{p}} < +\infty.
$$
Keywords:Lie ball, Shilov boundary, Fatou's theorem, hyperfuctions, parabolic subgroup, homogeneous line bundle Categories:32A45, 30E20, 33C67, 33C60, 33C55, 32A25, 33C75, 11K70 |
13. CJM 2010 (vol 62 pp. 889)
Singular Integral Operators and Essential Commutativity on the Sphere Let ${\mathcal T}$ be the $C^\ast $-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$.
Categories:32A55, 46L05, 47L80 |
14. CJM 2009 (vol 62 pp. 439)
On Hankel Forms of Higher Weights: The Case of Hardy Spaces In this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by SundhÃ¤ll for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.
Keywords:Hankel forms, Schattenâvon Neumann classes, Bergman spaces, Hardy spaces, Besov spaces, transvectant, unitary representations, MÃ¶bius group Categories:32A25, 32A35, 32A37, 47B35 |
15. CJM 2009 (vol 62 pp. 218)
The General Definition of the Complex Monge--AmpÃ¨re Operator on Compact KÃ¤hler Manifolds We introduce a wide subclass ${\mathcal F}(X,\omega)$ of
quasi-plurisubharmonic functions in a compact KÃ¤hler manifold, on
which the complex Monge-AmpÃ¨re operator is well defined and the
convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$
is a convex cone and includes all quasi-plurisubharmonic functions
that are in the Cegrell class.
Keywords:complex Monge--AmpÃ¨re operator, compact KÃ¤hler manifold Categories:32W20, 32Q15 |
16. CJM 2009 (vol 62 pp. 3)
On the Asymptotic Behavior of Complete KÃ¤hler Metrics of Positive Ricci Curvature Let $( X,g) $ be a complete noncompact KÃ¤hler manifold, of
dimension $n\geq2,$ with positive Ricci curvature and of standard type
(see the definition below). N. Mok proved that $X$ can be
compactified, \emph{i.e.,} $X$ is biholomorphic to a quasi-projective
variety$.$ The aim of this paper is to prove that the $L^{2}$
holomorphic sections of the line bundle $K_{X}^{-q}$ and the volume
form of the metric $g$ have no essential singularities near the
divisor at infinity. As a consequence we obtain a comparison between
the volume forms of the KÃ¤hler metric $g$ and of the Fubini--Study
metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we
establish a relation between the number of components of the divisor
$D$ and the dimension of the groups $H^{i}( \overline{X},
\Omega_{\overline{X}}^{1}( \log D) )$.
Categories:53C55, 32A10 |
17. CJM 2009 (vol 61 pp. 1407)
Traces, Cross-Ratios and 2-Generator Subgroups of $\SU(2,1)$ In this work, we investigate how to decompose a pair $(A,B)$ of
loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under
the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are
involutions. The main result is a decomposability criterion, which
is expressed in terms of traces of elements of the group $\langle
A,B\rangle$.
Categories:14L24, 22E40, 32M15, 51M10 |
18. CJM 2009 (vol 61 pp. 566)
Convex Subordination Chains in Several Complex Variables In this paper we study the notion of a convex subordination chain in several
complex variables. We obtain certain necessary and sufficient conditions for a
mapping to be a convex subordination chain, and we give various examples of
convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We
also obtain a sufficient condition for injectivity of $f(z/\|z\|,\|z\|)$
on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain
over $(0,1)$.
Keywords:biholomorphic mapping, convex mapping, convex subordination chain, Loewner chain, subordination Categories:32H02, 30C45 |
19. CJM 2009 (vol 61 pp. 50)
Composition operators on $\mu$-Bloch spaces Given a positive continuous function $\mu$ on the
interval $0 Categories:47B33, 32A70, 46E15 |
20. CJM 2008 (vol 60 pp. 1219)
CR Extension from Manifolds of Higher Type This paper deals with the extension of CR functions
from a manifold $M\subset \mathbb C^n$ into directions produced by higher
order commutators of holomorphic and antiholomorphic vector fields. It
uses the theory of complex ``sectors'' attached to real submanifolds
introduced in recent joint work of the authors with D. Zaitsev. In
addition, it develops a new technique of approximation of sectors by
smooth discs.
Categories:32V25, 32V35, 32C16, 32F18 |
21. CJM 2008 (vol 60 pp. 721)
Uniform Linear Bound in Chevalley's Lemma We obtain a uniform linear bound for the Chevalley function at a point in
the source of an analytic mapping that is regular in the sense of
Gabrielov. There is a version of
Chevalley's lemma also along a fibre, or at a point of the image of a proper
analytic mapping. We get a uniform linear bound for the Chevalley
function of a closed Nash (or formally Nash) subanalytic set.
Keywords:Chevalley function, regular mapping, Nash subanalytic set Categories:13J07, 32B20, 13J10, 32S10 |
22. CJM 2008 (vol 60 pp. 33)
Higher Order Tangents to Analytic Varieties along Curves. II Let~$V$ be an analytic variety in some open set in~$\C^n$. For a
real analytic curve~$\gamma$ with $ \gamma(0) = 0 $ and $ d \ge 1 $
define $ V_t = t^{-d}(V - \gamma(t)) $. It was shown in a previous
paper that the currents of integration over~$V_t$ converge to a
limit current whose support $ T_{\gamma,d} V $ is an algebraic
variety as~$t$ tends to zero. Here, it is shown that the canonical
defining function of the limit current is the suitably normalized
limit of the canonical defining functions of the~$V_t$. As a
corollary, it is shown that $ T_{\gamma,d} V $ is either
inhomogeneous or coincides with $ T_{\gamma,\delta} V $ for
all~$\delta$ in some neighborhood of~$d$. As another application it
is shown that for surfaces only a finite number of curves lead to
limit varieties that are interesting for the investigation of
Phragm\'en--Lindel\"of conditions. Corresponding results for limit
varieties $ T_{\sigma,\delta} W $ of algebraic varieties W along
real analytic curves tending to infinity are derived by a
reduction to the local case.
Category:32C25 |
23. CJM 2007 (vol 59 pp. 1121)
Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series Meromorphic continuation of the Eisenstein series induced from spherical,
cuspidal data on parabolic subgroups is achieved via reworking
Bernstein's adaptation of Selberg's third proof of meromorphic
continuation.
Categories:11F72, 32N10, 32D15 |
24. CJM 2007 (vol 59 pp. 1098)
Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions In this paper we study ruled surfaces which appear as an exceptional
surface in a succession of blowing-ups. In particular we prove
that the $e$-invariant of such a ruled exceptional surface $E$ is
strictly positive whenever its intersection with the other
exceptional surfaces does not contain a fiber (of $E$). This fact
immediately enables us to resolve an open problem concerning an
intersection configuration on such a ruled exceptional surface
consisting of three nonintersecting sections. In the second part
of the paper we apply the non-vanishing of $e$ to the study of the
poles of the well-known topological, Hodge and motivic zeta
functions.
Categories:14E15, 14J26, 14B05, 14J17, 32S45 |
25. CJM 2007 (vol 59 pp. 1069)
Quotients jacobiens : une approche algÃ©brique Le diagramme d'Eisenbud et Neumann d'un germe est un arbre qui
repr\'esente ce germe et permet d'en calculer les invariants. On donne
une d\'emonstration alg\'ebrique d'un r\'esultat caract\'erisant
l'ensemble des quotients jacobiens d'un germe d'application $(f,g)$
\`a partir du diagramme d'Eisenbud et Neumann de $fg$.
Keywords:SingularitÃ©, jacobien, quotient jacobien, polygone de Newton Categories:14B05, 32S05, 32S50 |