Mathematics - Complex Variables Publications (50)

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Mathematics - Complex Variables Publications

Let $\Omega$ be a pseudoconvex domain in $\mathbb C^n$ satisfying an $f$-property for some function $f$, we show that the Bergman metric associated to $\Omega$ has the lower bound $\tilde g(\delta_\Omega(z)^{-1})$ where $\delta_\Omega(z)$ is the distance from $z$ to $\partial\Omega$ and $\tilde g$ is a specific function defined by $f$. This refines Khanh-Zampieri's work in \cite{KZ12} with reducing the smoothness assumption of the boundary. Read More


We prove almost sure invariance principle, a strong form of approximation by Brownian motion, for non-autonomous holomorphic dynamical systems on complex projective space $\Bbb{P}^k$ for H\"older continuous and DSH observables. Read More


We propose a new class of geometric invariants called jet vanishing orders, and use them to establish a new selection algorithm for generators of the Kohn's multiplier ideals for special domains in dimension $3$. In particular, we obtain the effective termination of our selection algorithm with explicit bounds. Read More


In this paper, we consider a general subclass of analytic and bi-univalent functions in the open unit disk in the complex plane. Making use of the Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant for this function class. Read More


We prove a strong version of the comparison principle for bounded plurisubharmonic function on complex varieties. we then apply our main result to study convergence of Mong-Ampere mesures for bounded plurisubharmonic functions. Read More


We give upper and lower bounds for the number of solutions of the equation $p(z)\log|z|+q(z)=0$ with polynomials $p$ and $q$. Read More


This paper is devoted to study multiplicity and regularity as well as to present some classifications of complex analytic sets. We present an equivalence for complex analytical sets, namely blow-spherical equivalence and we receive several applications with this new approach. For example, we reduce to homogeneous complex algebraic sets a version of Zariski's multiplicity conjecture in the case of blow-spherical homeomorphism, we give some partial answers to the Zariski's multiplicity conjecture, we show that a blow-spherical regular complex analytic set is smooth and we give a complete classification of complex analytic curves. Read More


We show that the Rudin-Shapiro polynomials $P_n$ and $Q_n$ of degree $N-1$ with $N := 2^n$ have $o(N)$ zeros on the unit circle. This should be compared with a result of B. Conrey, A. Read More


We present several upper bounds for the height of global residues of rational forms on an affine variety. As a consequence, we deduce upper bounds for the height of the coefficients in the Bergman-Weil trace formula. We also present upper bounds for the degree and the height of the polynomials in the elimination theorem on an affine variety. Read More


The space of K\"ahler potentials in a compact K\"ahler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of K\"ahler potentials, and prove existence and uniqueness for such isometries. Read More


The aim of this research paper is to obtain explicit expressions of (i) $ {}_1F_1 \left[\begin{array}{c} \alpha \\ 2\alpha + i \end{array} ; x \right]. {}_1F_1\left[ \begin{array}{c} \beta \\ 2\beta + j \end{array} ; x \right]$ (ii) ${}_1F_1 \left[ \begin{array}{c} \alpha \\ 2\alpha - i \end{array} ; x \right] . {}_1F_1 \left[ \begin{array}{c} \beta \\ 2\beta - j \end{array} ; x \right]$ (iii) ${}_1F_1 \left[ \begin{array}{c} \alpha \\ 2\alpha + i \end{array} ; x \right] . Read More


New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Read More


We show that there is an absolute constant $c > 1/2$ such that the Mahler measure of the Fekete polynomials $f_p$ of the form $$f_p(z) := \sum_{k=1}^{p-1}{\left( \frac kp \right)z^k}\,,$$ (where the coefficients are the usual Legendre symbols) is at least $c\sqrt{p}$ for all sufficiently large primes $p$. This improves the lower bound $\left(\frac 12 - \varepsilon\right)\sqrt{p}$ known before for the Mahler measure of the Fekete polynomials $f_p$ for all sufficiently large primes $p \geq c_{\varepsilon}$. Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle. Read More


Given a polyanalytic function, we show that the corresponding Toeplitz operator on the Bergman space of the unit disc can be expressed as a quotient of certain differential operators with holomorphic coefficients. This enables us to obtain several operator theoretic results including a criterion for invertibility of a Toeplitz operator. Read More


In this paper, we prove the $C^{1, 1}$-regularity of the plurisubharmonic envelope of a $C^{1,1}$ function on a compact Hermitian manifold. We also present examples to show this regularity is sharp. Read More


We consider two-variable model spaces associated to rational inner functions $\Theta$ on the bidisk, which always possess canonical $z_2$-invariant subspaces $\mathcal{S}_2.$ A particularly interesting compression of the shift is the compression of multiplication by $z_1$ to $\mathcal{S}_2$, namely $ S^1_{\Theta}:= P_{\mathcal{S}_2} M_{z_1} |_{\mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with well-behaved symbols and characterize their numerical ranges and radii. Read More


We establish the C^{1,1} regularity of quasi-psh envelopes in a Kahler class, confirming a conjecture of Berman. Read More


Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a compact Lie group action $G$. Under certain natural assumptions about the group action $G$, we show that the Szeg\"o kernel for $(0,q)$ forms is a complex Fourier integral operator, smoothing away $\mu^{-1}(0)$ and there is a precise description of the singularity near $\mu^{-1}(0)$, where $\mu$ denotes the CR moment map. Read More


We show that a discrete sequence $\Lambda$ of the unit disk is the union of $n$ interpolating sequences for the Nevanlinna class $N$ if and only if the trace of $N$ on $\Lambda$ coincides with the space of functions on $\Lambda$ for which the divided differences of order $n - 1$ are uniformly controlled by a positive harmonic function. Read More


Recently Mereknov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpi\'nski carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpi\'nski carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova. Read More


In the present investigation, by applying two different normalizations of the Jackson and Hahn-Exton $q$-Bessel functions tight lower and upper bounds for the radii of convexity of the same functions are obtained. In addition, it was shown that these radii obtained are solutions of some transcendental equations. The known Euler-Rayleigh inequalities are intensively used in the proof of main results. Read More


We prove that a $IR n+1$-valued vector field on IR n is the sum of the traces of two harmonic gradients, one in each component of $IR n+1 \ IR n$ , and of a $IR n$-valued divergence free vector field. We apply this to the description of vanishing potentials in divergence form. The results are stated in terms of Clifford Hardy spaces, the structure of which is important for our study. Read More


We study the change in the extrinsic geometry of a triangulated surface under infinitesimal conformal deformations in Euclidean space. A deformation of vertices is conformal if it preserves length cross-ratios. On the one hand, conformal deformations generalize deformations preserving edge lengths. Read More


A theorem of Kuranishi tells us that the moduli space of complex structures on any smooth compact manifold is always locally a finite-dimensional space. Globally, however, this is simply not true; we display examples in which the moduli space contains a sequence of regions for which the local dimension tends to infinity. These examples naturally arise from the twistor theory of hyper-Kaehler manifolds. Read More


We prove several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, we prove a generalization of the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles on toric varieties. Read More


A Riemannian metric on a compact 4-manifold is said to be Bach-flat if it is a critical point for the L2-norm of the Weyl curvature. When the Riemannian 4-manifold in question is a Kaehler surface, we provide a rough classification of solutions, followed by detailed results regarding each case in the classification. The most mysterious case prominently involves 3-dimensional CR manifolds. Read More


The origin of quasiconformal mappings, like that of conformal mappings, can be traced back to old cartography where the basic problem was the search for mappings from the sphere onto the plane with minimal deviation from conformality, subject to certain conditions which were made precise. In this paper, we survey the development of cartography, highlighting the main ideas that are related to quasiconformality. Some of these ideas were completely ignored in the previous historical surveys on quasiconformal mappings. Read More


A proof is reconstructed for a useful theorem on the zeros of derivatives of analytic functions due to H. M. Macdonald, which appears to be now little known. Read More


We study bounds for the backward shift operator $f \mapsto (f(z)-f(0))/z$ and the related operator $f \mapsto f - f(0)$ on Hardy and Bergman spaces of analytic and harmonic functions. If $u$ is a real valued harmonic function, we also find a sharp bound on $M_1(r,u-u(0))$ in terms of $\|u\|_{h^1}$, where $M_1$ is the integral mean with $p=1$. Read More


We study weighted boundedness of Hardy-Littlewood-type maximal function involving Orlicz functions. We also obtain some sufficient conditions for the weighted boundedness of the Hardy-Littlewood maximal function of the upper-half plane. Read More


We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space. Read More


In this paper, we prove that every real analytic totally nondegenerate model CR manifold of length >= 3 has rigidity. This result was actually conjectured before by Valerii Beloshapka as the so-called "maximum conjecture". It follows that the transformation Lie group of all CR automorphisms associated with each of the mentioned models does not include any nonlinear map. Read More


We prove that a meromorphic mapping, which sends a peace of a real analytic strictly pseudoconvex hypersurface in $\cc^2$ to a compact subset of $\cc^N$ which doesn't contain germs of non-constant complex curves is continuous from the concave side of the hypersurface. This implies the analytic continuability along CR-paths of germs of holomorphic mappings from real analytic hypersurfaces with non-vanishing Levi form to the locally spherical ones in all dimensions. Read More


Let Z be the zero set of a holomorphic map f from C^n to C^k with f(0) = 0. We prove that for any r > 0, the Gaussian measure of the Euclidean r-neighborhood of Z is at least as large as the Gaussian measure of the Euclidean r-neighborhood of an (n-k)-dimensional complex subspace of C^n. Read More


Let $\mathscr T=(V, \mathcal E)$ be a leafless, locally finite rooted directed tree. We associate with $\mathscr T$ a one parameter family of Dirichlet spaces $\mathscr H_q~(q \geqslant 1)$, which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc $\mathbb D$ in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel \begin{eqnarray*} \kappa_{\mathscr H_q}(z, w) = \sum_{n=0}^{\infty}\frac{(1)_n}{(q)_n}\,{z^n \overline{w}^n} ~P_{\langle e_{\mathsf{root}}\rangle} + \sum_{v \in V_{\prec}} \sum_{n=0}^{\infty} \frac{(n_v +2)_n}{(n_v + q+1)_n}\, {z^n \overline{w}^n}~P_{v}~(z, w \in \mathbb D), \end{eqnarray*} where $V_{\prec}$ denotes the set of branching vertices of $\mathscr T$, $n_v$ denotes the depth of $v \in V$ in $\mathscr T,$ and $P_{\langle e_{\mathsf{root}}\rangle}$, $~P_{v}~(v \in V_{\prec})$ are certain orthogonal projections. Read More


We show that the Diederich-Fornaess index of a domain in a Stein manifold is invariant under CR-diffeomorphisms. For this purpose we also improve CR-extension theorem. Read More


We study the family of holomorphic maps from the polydisk to the disk which restrict to the identity on the diagonal. In particular, we analyze the asymptotics of the orbit of such a map under the conjugation action of a unipotent subgroup of $\text{PSL}_2(\mathbb{R})$. We discuss an application our results to the study of the Carath\'eodory metric on Teichm\"uller space. Read More


In this paper we show that if $p$ is a polynomial which bifurcates then the product map $(z,w)\mapsto(p(z),q(w))$ can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets can be chosen to be of two types : repelling or saddle. As a consequence, such product map belongs to the closure of the interior of two different sets : the bifurcation locus of $H_d(\mathbb P^2)$ and the set of endomorphisms having an attracting set of non-empty interior. Read More


Using generalized hypergeometric functions to perform symbolic manipulation of equations is of great importance to pure and applied scientists. There are in the literature a great number of identities for the Meijer-G function. On the other hand, when more complex expressions arise, the latter function is not capable of representing them. Read More


We study general properties of holomorphic isometric embeddings of complex unit balls $\mathbb B^n$ into bounded symmetric domains of rank $\ge 2$. In the first part, we study holomorphic isometries from $(\mathbb B^n,kg_{\mathbb B^n})$ to $(\Omega,g_\Omega)$ with non-minimal isometric constants $k$ for any irreducible bounded symmetric domain $\Omega$ of rank $\ge 2$, where $g_D$ denotes the canonical K\"ahler-Einstein metric on any irreducible bounded symmetric domain $D$ normalized so that minimal disks of $D$ are of constant Gaussian curvature $-2$. In particular, results concerning the upper bound of the dimension of isometrically embedded $\mathbb B^n$ in $\Omega$ and the structure of the images of such holomorphic isometries were obtained. Read More


Behavior of solutions of $f''+Af=0$ is discussed under the assumption that $A$ is analytic in $\mathbb{D}$ and $\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature. Read More


This paper is devoted to the uniqueness problem of the power of a meromorphic function with its differential polynomial sharing a set. Our result will extend a number of results obtained in the theory of normal families. Some questions are posed for future research. Read More


We prove that the rank of a non-trivial co-doubly commuting submodule is $2$. More precisely, let $\varphi, \psi \in H^\infty(\mathbb{D})$ be two inner functions. If $\mathcal{Q}_{\varphi} = H^2(\mathbb{D})/ \varphi H^2(\mathbb{D})$ and $\mathcal{Q}_{\psi} = H^2(\mathbb{D})/ \psi H^2(\mathbb{D})$, then \[ \mbox{rank~}(\mathcal{Q}_{\varphi} \otimes \mathcal{Q}_{\psi})^\perp = 2. Read More


In this paper we give a new proof, relying on Banach's contraction mapping principle, of a celebrated theorem of Andr\'e Bloch. Also, via the same contraction mapping principle, we give a proof of a Bloch type theorem for normalised Wu $K$-mappings. Read More


We embed compact $C^\infty$ manifolds into $\mathbb C^n$ as totally real manifolds with prescribed polynomial hulls. As a consequence we show that any compact $C^\infty$ manifold of dimension $d$ admits a totally real embedding into $\mathbb C^{\lfloor \frac{3d}{2}\rfloor}$ with non-trivial polynomial hull without complex structure. Read More


An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theorem and Montel theorem. Read More


In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions. Read More


Consider a holomorphic foliation with singularities of a 2-dimensional complex manifold. In this article we prove a new sufficient condition for this foliation to have countably many homologically independent complex limit cycles. In particular, if all leaves of a foliation are dense in the phase space, and it has a complex hyperbolic singular point, then it has infinitely many homologically independent complex limit cycles. Read More


We use a special version of the Corona Theorem in several variables, valid when all but one of the data functions are smooth, to generalize to the polydisc results obtained by El Fallah, Kellay and Seip about cyclicity of non vanishing bounded holomorphic functions in large enough Banach spaces of analytic functions on the polydisc determined either by weighted sums of powers of Taylor coefficients or by radially weighted integrals of powers of the modulus of the function. Read More


In this paper we find a holomorphic Darboux chart along any immersed noncompact holomorphic Legendrian curve in a contact complex manifold $(X,\xi)$. As an application we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated, uniformly on any relatively compact subset $U\Subset R$, by holomorphic Legendrian embeddings $U\hookrightarrow X$. We also prove that every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. Read More