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A complete characterization of proper holomorphic mappings between domains from the class of all pseudoconvex Reinhardt domains in $\C^2$ with the logarithmic image equal to a strip or a half-plane is given. |
In this paper we sharpen significantly several known estimates on the maximal number of zeros of complex harmonic polynomials. We also study the relation between the curvature of critical lemniscates and its impact on geometry of caustics and the number of zeros of harmonic polynomials. |
We prove that any diffeomorphism of the sphere S^n to itself can be decomposed into bi-Lipschitz mappings of small isometric distortion and which move points a small amount in the spherical metric. |
In this paper we prove a general result of the ``Hopf lemma'' type for CR mappings, with nonidentically vanishing Jacobians, between real hypersurfaces in C^n with smooth or real analytic boundaries. Applications of this result to finiteness and holomorphic extendibility of such mappings are also given. |
We consider versions of the local duality theorem in $\mathbb{C}^n$. We show that there exist canonical pairings in these versions of the duality theorem which can be expressed explicitly in terms of residues of Grothendieck, or in terms of residue currents of Coleff-Herrera and Andersson-Wulcan, and we give several d... |
We investigate the frame set of regular multivariate Gaussian Gabor frames using methods from Kähler geometry such as Hörmander's $\dbar$-$L^2$ estimate with singular weight, Demailly's Calabi--Yau method for Kähler currents and a Kähler-variant generalization of the symplectic embedding theorem of McDuff--Polt... |
Using Ahlfors functions, Grunsky maps and the Bell representation theorem, we show that a certain subset of the rational maps of degree $n$ forms a trivial bundle over the moduli space of non-degenerate $n$-connected domains with one marked tangent vector with fiber the $n$-fold symmetric product of the circle. A cons... |
We explain how, given a plane algebraic curve $\mathcal{C}\colon f(x,y) = 0$, $x_1 \in \mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\varepsilon > 0$, we can compute $\delta > 0$ such that $|y_j(x_1) - y_j(x_2)| < \varepsilon$ for all holomorphic functions $y_j(x)$ which satisfy $f(x, y_j(x)) = 0$ in a... |
Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve $C$ with positive self-intersection. We prove that there exists a neighborhood $U\supset C$ such that any meromorphic function defined on a connected neighborhood of $C$ in $U$ can be extended to a m... |
The classical results about the boundary values of holomorphic or harmonic functions on a domain $D$ state that under additional integrability assumptions these functions have limits along specific sets approaching boundary. The proofs of these results are based on properties of smooth boundaries used to define the ap... |
We characterize the Schatten class membership of the canonical solution operator to $\bar\partial$ acting on $L^2(e^{-2\phi})$, where $\phi$ is a subharmonic function with $\Delta\phi$ a doubling measure. The obtained characterization is in terms of $\Delta\phi$. |
We propose a conjecture that a general member of a bracket-generating family of rational curves in a complex manifold satisfies the formal principle with convergence, namely, any formal equivalence between such curves is convergent. If the normal bundles of the rational curves are positive, the conjecture follows from... |
In this paper, we expand functions of specific $q$-exponential growth in terms of its even (odd) Askey- Wilson $q$-derivatives at $0$ and $\eta=(q^{1/4}+q^{-1/4})/2$. This expansion is a $q$-version of the celebrated Lidstone expansion theorem, where we expand the function in $q$-analogs of Lidstone polynomials, i.e.,... |
The main goal of these notes, compiled in 2008-2009, is to present a more-or-less self-contained discussion of some of the recent results and techniques of R. Berman and S. Boucksom in the setting of weighted pluripotential theory. We include some background results on pluripotential theory and weighted pluripotential... |
Hayman's conjecture about covering of vertical intervals under regular mappings is proved. |
This paper shows that an arbitrary generic submanifold in a complex manifold can be deformed into a 1-parameter family of generic submanifolds satisfying strong nondegeneracy conditions. The proofs use a careful analysis of the jet spaces of embeddings satisfying certain nondegeneracy properties, and also make use of ... |
We prove that if two real-analytic hypersurfaces in $\mathbb C^2$ are equivalent formally, then they are also $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (in particular are conve... |
In the present paper we answer a question raised by J. Borcea and P. Branden and give a description of the class of operators preserving roots in open circular domains, i.e., in images of the open upper half-plane under the Mobius transformations. Our second result is a description of the difference between $\Apr(G)$ ... |
We introduce the symplectic holomorphic density property and the Hamiltonian holomorphic density property together with the corresponding version of Andersén-Lempert theory. We establish these properties for the Calogero-Moser space $\mathcal{C}_n$ of $n$ particles and describe its group of holomorphic symplectic auto... |
A group $H \cong {\mathbb Z}_{k}^{2g}$, where $g,k \geq 2$ are integers, of conformal automorphisms of a closed Riemann surface $S$ is called a $(g,k)$-Fermat group if it acts freely with quotient $S/H$ of genus $g$. We study some properties of these type of objects, in particular, we observe that $S$ is non-hyperelli... |
We prove that the upper envelope of a family of subharmonic functions defined on an open subset of $\mathbb{R}^{N}$, $(N\geq2)$, that is finite every where, is locally bounded above outside a closed nowhere dense set with no bounded components. Then we conclude as a consequence that a separately subharmonic function i... |
We obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. |
We survey a number of recent generalizations and sharpenings of Nehari's extension of Schwarz' lemma for holomorphic self-maps of the unit disk. In particular, we discuss the case of infinitely many critical points and its relation to the zero sets and invariant subspaces for Bergman spaces, as well as the cas... |
Focus in this paper is on the Hankel determinant, $H_3(1)$, for the well-known classes of bounded-turning, starlike and convex functions in the open unit disk $E=\{z\in \mathbb{C}\colon|z|<1\}$. The results obtained complete the series of research works in the search for sharp upper bounds on $H_3(1)$ for each of t... |
In this article we prove Bohr inequalities for sense-preserving $K$-quasiconformal harmonic mappings defined in $\mathbb{D}$ and obtain the corresponding results for sense-preserving harmonic mappings by letting $K\to\infty$. One of the results includes the sharpened version of a theorem by Kayumov $\textit{et. |
The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and gen... |
We study an equation proposed by Fu and Yau as a natural $n$-dimensional generalization of a Strominger system that they solved in dimension $2$. It is a complex Hessian equation with right hand side depending on gradients. |
Early in 1970s, Carlson-Griffiths made a significant progress in the study of Nevanlinna theory, who devised equi-distribution theory for holomorphic mappings from $\mathbb C^m$ into a projective algebraic manifold intersecting divisors. In this paper, we develop Carlson-Griffiths' theory by generalizing the sourc... |
We give an overview of some recent developments concerning harmonic and other moments of plane domains, their relationship to the Cauchy and exponential transforms, and to the meromorphic resultant and elimination function. The paper also connects to certain topics in mathematical physics, for example domain deformati... |
This is the second paper in a series of investigations of the pluripotential theory on Teichmüller space. The main purpose of this paper is to establish the Poisson integral formula for pluriharmonic functions on Teichmüller space which are continuous on the Bers compactification. |
In the present investigation, we introduce and study the geometric properties of a class of analytic functions, associated with a parabolic region majorly lying in the left-half plane. Further we establish radius and majorization results for the class under study with pictorial illustrations of some of its special cas... |
Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implic... |
Given a compact Kähler manifold $X$, a quasiplurisubharmonic function is called a Green function with pole at $p\in X$ if its Monge-Ampère measure is supported at $p$. We study in this paper the existence and properties of such functions, in connection to their singularity at $p$. |
As a continuation of our previous work \cite{KV2} the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions... |
Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ We show that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of $\mathcal{F}_d(\mathbb{P}^n)$ with full Lebesgue measure with the following properties: <br>1. for every $... |
In this paper we fully describe functions generating the infinite totally nonnegative Hurwitz matrices. In particular, we generalize the well-known result by Asner and Kemperman on the total nonnegativity of the Hurwitz matrices of real stable polynomials. |
We extend results on compressed Toeplitz operators on the backward shift invariant subspaces of $H^2 $ to the context of the spaces $H^p$, $1<p<\infty. $ |
We study the structure of a class of laminar closed positive currents on $\mathbb{CP}^2$, naturally appearing in birational dynamics. We prove such a current admits natural non intersecting {\em leaves}, that are closed under analytic continuation. |
In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes t... |
We derive a useful result about the zeros of the $k$-polar polynomials on the unit circle; in particular we obtain a ring shaped region containing all the zeros of these polynomials. Some examples are presented. |
Let $n$ be a positive integer and $\mathbf{g}=(g_0,g_1,\cdots,g_{n-1})$, with $g_k\in H(\mathbb{D})$ for $k=0,1,\cdots,n-1$. Let $I_{\mathbf{g}}^{(n)}$ be the generalized Volterra-type operators on $H(\mathbb{C})$, which is represented as $$ I_{\mathbf{g}}^{(n)}f=I^n\left(fg_0+f'g_1+\cdots+f^{(n-1)}g_{n-1}\right),... |
For $f$ analytic and close to convex in $D=\{z: |z|< 1\}$, we give sharp estimates for the logarithmic coefficients $\gamma_{n}$ of $f$ defined by $\log \dfrac{f(z)}{z}=2\sum_{n=1}^{\infty} \gamma_{n}z^{n}$ when $n=1, 2,3$. |
The configuration space C^n of unordered n-tuples of distinct points on a torus T^2 is a non-singular complex algebraic variety. We study holomorphic self-maps of C^n and prove that for n>4 any such map F either carries the whole of C^n into an orbit of the diagonal Aut(T^2) action in C^n or is of the form F(x)=T(x... |
We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. |
In this paper, we first study the bounded mean oscillation of planar harmonic mappings, then a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings is established. At last, we obtain sharp estimates on Lipschitz number of planar harmonic mappings in terms of bounded mean oscillat... |
It has been conjectured that every stable manifold arising from a holomorphic automorphism, that acts hyperbolically on a compact invariant set, is biholomorphic to complex Euclidean space. Such stable manifolds are known to be biholomorphic to the basin of a uniformly attracting family of holomorphic maps. |
The purpose of this work is to introduce the reader to the Woods Hole trace formula and to show how several well-known index theorems for foliations, vector fields and endomorphisms can be rephrased as particular cases of such formula. The main objective in doing this is to use the Woods Hole formula in the future to ... |
If $f=u+iv$ is analytic in the unit disk $\mathbb{D}$, it is known that the integral means $M_p(r,u)$ and $M_p(r,v)$ have the same order of growth. This is false if $f$ is a (complex-valued) harmonic function. |
Class I CR manifolds have initial G-structure a certain 4-dimensional subgroup of GL_3(C). Class II CR manifolds have initial G-structure a certain 10-dimensional subgroup of GL_4(C). |
Let $X$ be a compact Kähler manifold and $(L,h)\rightarrow X$ be a pseudoeffective line bundle, such that the curvature $i\Theta_{L,h}\geq 0$ in the sense of currents. The main result of the present paper is that $H^n(X,\mathcal{O}(\Omega^p_X\otimes L)\otimes \mathcal{I}(h))=0$ for $p\geq n-nd(L,h)+1$. |
We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the first author. |
Let $\alpha$ \in (0; 1). We show that any $\alpha$-Hölder homeomorphism from the unit circle in the plane to the plane can be extended to an $\alpha$-Hölder homeomorphism from the whole unit disc. |
We study Lelong numbers of currents of full mass intersection on a compact Kaehler manifold in a mixed setting. Our main theorems cover some recent results due to Darvas-Di Nezza-Lu. |
We discuss the zero sets of two-variable polynomials as they relate to an approximation problem in the Hardy space on the bidisk. |
It is shown that the weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $C^{1+\varepsilon_-}$-smooth boundary. On the other hand, it is proved that the weak converse to the Suita conjecture holds for any finitely connected planar domain. |
The functorial mathematical definition of conformal field theory was first formulated approximately 30 years ago. The underlying geometric category is based on the moduli space of Riemann surfaces with parametrized boundary components and the sewing operation. |
The well-known Axler-Zheng theorem characterizes compactness of finite sums of finite products of Toeplitz operators on the unit disk in terms of the Berezin transform of these operators. Subsequently this theorem was generalized to other domains and appeared in different forms, including domains in $\mathbb{C}^n$ on ... |
We introduce mixed twistor $D$-modules, and establish the fundamental functorial property. We also prove that they are described as the gluing of admissible variations of mixed twistor structure. |
We study the relationship between sampling sequences in infinite dimensional Hilbert spaces of analytic functions and Marcinkiewicz-Zygmund inequalities in subspaces of polynomials. We focus on the study of the Hardy space and the Bergman space in one variable because they provide two settings with a strikingly differ... |
In this work we show how the Multiplicity Polar Theorem can be used to calculate Chern numbers for a collection of 1-forms. |
In this article, we have introdued D-fuzzy sets. We have discussed the notions of inclusion, union, intersection, complementation and convexity of such D-fuzzy sets. |
We say that a class $\mathcal{B}$ of analytic functions $f$ of the form $f(z)=\sum_{n=0}^{\infty} a_{n}z^{n}$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}: |z|<1\}$ satisfies a Bohr phenomenon if for the largest radius $R_{f}<1$, the following inequality $$ \sum\limits_{n=1}^{\infty} |a_{n}z^{n}| \leq d(f(0),\... |
In this paper, we solve certain Fermat-type partial differential-difference equations for finite order entire functions of several complex variables. These results are significant generalizations of some earlier findings, especially those of Haldar and Ahamed (Entire solutions of several quadratic binomial and trinomi... |
We give in this note a generalisation of the following theorem of Henkin and Passare : Let be Y an analytic subvariety of pure codimension p in a linearly p-concave domain U, and w a meromorphic q-form (q>0) on Y; if the Abel transform of w, which is meromorphic on U*, has a meromorphic prolongation to some greater ... |
Let $X$ be a Stein manifold of complex dimension $n>1$ endowed with a Riemannian metric $\mathfrak{g}$. We show that for every integer $k$ with $\left[\frac{n}{2}\right] \le k \le n-1$ there is a nonsingular holomorphic foliation of dimension $k$ on $X$ all of whose leaves are topologically closed and $\mathfrak{g}... |
Let $0<p<\infty$, $\beta>-1$, and $\Omega$ be a strongly pseudoconvex bounded domain with a smooth boundary in $\mathbb{C}^n$. We will study the interpolation problem for weighted Bergman spaces $A^p_\beta(\Omega)$. |
For a given $\epsilon >0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+\epsilon)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $\epsilon>0$ there are two finite regular covers of the Modular once punctured... |
We look for pointwise bounds on a plurisubharmonic function near its singularity point, given the value of its generalized Lelong number with respect to a plurisubharmonic weight. To this end, an extremal problem is considered. |
We give an elementary characterization of rational functions among meromorphic functions in the complex plane. |
Let $D$ be a bounded domain in $\mathbf C^2$ with a non-compact group of holomorphic automorphisms. Model domains for $D$ are obtained under the hypothesis that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. |
In this article, we characterize plurisubharmonic functions of Lelong number one at the origin, such that the germ of the associated multiplier ideal sheaf is nontrivial: in arbitrary complex dimension, their singularity must be the sum of a germ of smooth divisor and of a plurisubharmonic function with zero Lelong num... |
We provide a direct construction of Poletsky discs via local arc approximation and a Runge-type theorem by A. Gournay. |
In this note, we show how to apply the original $L^2$-extension theorem of Ohsawa and Takegoshi to the standard basis of a multiplier ideal sheaf associated with a plurisubharmonic function. In this way, we are able to reprove the strong openness conjecture and to obtain an effective version of the semicontinuity theo... |
The purpose of this article is to study Lipschitz CR mappings from an $h$-extendible (or semi-regular) hypersurface in $\mbb C^n$. Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. |
This paper addresses particular eigenvalue problems within the context of two quaternionic function theories. More precisely, we study two concrete classes of quaternionic eigenvalue problems, the first one for the slice derivative operator in the class of quaternionic slice-regular functions and the second one for th... |
Using recent results of Berman and Boucksom we show that for a non-pluripolar compact set K in C^d and an admissible weight function w=e^{-\phi} any sequence of so-called optimal measures converges weak-* to the equilibrium measure \mu_{K,\phi} of (weighted) Pluripotential Theory for K,\phi. |
This is a revised version of the second half of my paper <a href="https://arxiv.org/abs/math.AG/9909021" data-arxiv-id="math.AG/9909021" class="link-https">math.AG/9909021</a>. We prove that there exists a positive integer $\nu_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $... |
Many classical objects on a surface S can be interpreted as cross-ratio functions on the circle at infinity of the universal covering. This includes closed curves considered up to homotopy, metrics of negative curvature considered up to isotopy and, in the case of interest here, tangent vectors to the Teichmüller spac... |
We study the asymptotic behavior of the ratio $|f(z)|/|z|$ as $z\to 0$ for mappings differentiable a.e. in the unit disc with non-degenerated Jacobian. The main tools involve the length-area functionals and angular dilatations depending on some real number $p. |
In this article the bicomplex version of Mobius transformation is defined and special attention is paid to find the fixed points of a bicomplex Mobius transformation. |
In this paper we study the function algebra generated by z^2 and g^2 on a small closed disk centered at the origin of the complex plane. We prove, using a biholomorphic change of coordinates and already developed techniques in this area, that for a large class of functions g this algebra consists of all continuous fun... |
We study classes of locally biholomorphic mappings defined in the $¶$ that have bounded Schwarzian operator in the Bergman metric. We establish important properties of specific solutions of the associated system of differential equations and show a geometric connection between the order of the classes and a covering p... |
We introduce a general multisummability theory of formal power series in Carleman ultraholomorphic classes. The finitely many levels of summation are determined by pairwise comparable, nonequivalent weight sequences admitting nonzero proximate orders and whose growth indices are distinct. |
This paper presents a primitive $n$th root of unity in $\mathbb C$. The approach is very elementary and avoids the following: the complex exponential function, trigonometry, and group theory. |
We develop the theory of the lemniscatic functions sl and cl from their definition as solutions to an initial value problem. |
Comparing phase plots of truncated series solutions of Kepler's equation by Lagrange's power series with those by Bessel's Kapteyn series strongly suggest that a Jentzsch-type theorem holds true not only for the former but also for the latter series: each point of the boundary of the domain of convergence i... |
We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. |
This note is a survey of Analysis on Metric spaces, in connection with the upcoming AMS Mathematics Research Communities program in June 2020. |
We give a concrete sufficient condition for a simply-connected domain to be the image of the unit disk under a nonexpansive conformal map. This class of domains is also characterized by having sufficiently dense harmonic measure. |
This article used Bloch function to derive Schottky inequality, obtained its generalization by using elliptic integral deviation function and demonstrated its applications. |
The purpose of this paper is to study the properties of the solutions to the biharmonic equations: $\Delta(\Delta f)=g$, where $g:$ $\overline{\mathbb{D}}\rightarrow\mathbb{C}$ is a continuous function and $\overline{\mathbb{D}}$ denotes the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$. In f... |
We describe the asymptotic behavior of the mapping function at an analytic cusp compared with Kaiser's results for cusps with small perturbation of angles and the known explicit formulae for cusps with circular boundary curves. We propose a boundary curve parametrization by generalized power series which allows us... |
We study codimension $q \geq 2$ holomorphic foliations defined in a neighborhood of a point $P$ of a complex manifold that are completely integrable, i.e. with $q$ independent meromorphic first integrals. We show that either $P$ is a regular point, a non-isolated singularity or there are infinitely many invariant anal... |
The Lempert function for several poles $a_0, ..., a_N$ in a domain $\Omega$ of $\mathbb C^n$ is defined at the point $z \in \Omega$ as the infimum of $\sum^N_{j=0} \log|\zeta_j|$ over all the choices of points $\zeta_j$ in the unit disk so that one can find a holomorphic mapping from the disk to the domain $\Omega$ sen... |
In this article, we first establish operator-valued analogues of multidimensional refined Bohr inequality. Then we establish operator-valued analogues of multidimensional improved Bohr inequality with a certain power of the norm of the initial coefficient. |
Let $M= \Gamma \setminus \mathbb{H}_d$ be a compact quotient of the $d$-dimensional Heisenberg group $\mathbb{H}_d$ by a lattice subgroup $\Gamma$. We show that the eigenvalue counting function $N(\lambda)$ for any fixed element of a family of second order differential operators $\left\{\mathcal{L}_\alpha\right\}$ on ... |
We present results about spaces of holomorphic functions associated to the classical Dirichlet space. The spaces we consider have roles similar to the roles of $H^{1}$ and $BMO$ in the Hardy space theory and we emphasize those analogies. |
It is shown that the optimal upper and lower bounds for the Kobayashi distance near $\mathcal C^{2,\alpha}$-smooth strongly pseudoconvex boundary points obtained in L. Kosinski, N. Nikolov, A.Y. Okten: "Precise estimates of invariant distances on strongly pseudoconvex domains", Adv. Math. |
We study the projective logarithmic potential $\mathbb{G}_{\mu}$ of a Probability measure $\mu$ on the complex projective space $\mathbb{P}^{n}$. We prove that the Range of the operator $\mu\longrightarrow \mathbb{G}_{\mu}$ is contained in the (local) domain of definition of the complex Monge-Ampère operator acting on... |
In this paper we study the following "slice rigidity property": given two Kobayashi complete hyperbolic manifolds $M, N$ and a collection of complex geodesics $\mathcal F$ of $M$, when is it true that every holomorphic map $F:M\to N$ which maps isometrically every complex geodesic of $\mathcal F$ onto a complex... |
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