math.CV

Okounkov bodies and transfinite diameter

We present an explicit calculation of an Okounkov body associated to an algebraic variety. This is used to derive a formula for transfinite diameter on the variety. We relate this formula to a recent result of D. Witt Nystrom.

On Infinitesimal Strebel Points

In this paper, we prove that if $X$ is a Riemann surface of infinite analytic type and $[μ]_T$ is any element of Teichmüller space, then there exists $μ_1 \in [μ]_T$ so that $μ_1$ is an infinitesimal Strebel point.

A fast numerical method for ideal fluid flow in domains with multiple stirrers

A collection of arbitrarily-shaped solid objects, each moving at a constant speed, can be used to mix or stir ideal fluid, and can give rise to interesting flow patterns. Assuming these systems of fluid stirrers are two-dimensional, the mathematical problem of resolving the flow field - given a particular distribution of any finite number of stirrers of specified shape and speed - can be formulated as a Riemann-Hilbert problem. We show that this Riemann-Hilbert problem can be solved numerically using a fast and accurate algorithm for any finite number of stirrers based around a boundary integral equation with the generalized Neumann kernel. Various systems of fluid stirrers are considered, and our numerical scheme is shown to handle highly multiply connected domains (i.e. systems of many fluid stirrers) with minimal computational expense.

Sparse Beltrami coefficients, integral means of conformal mappings and the Feynman-Kac formula

In this note, we give an estimate for the dimension of the image of the unit circle under a quasiconformal mapping whose dilatation has small support. We also prove an analogous estimate for the rate of growth of a solution of a second-order parabolic equation given by the Feynman-Kac formula (with a sparsely supported potential) and introduce a dictionary between the two settings.

Integral Kahler Invariants and the Bergman kernel asymptotics for line bundles

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one forms, which do not contribute to the integral. We apply this decomposition formula to describe the asymptotic expansion of the Bergman kernel for positive line bundles and to show that the CR Q-curvature on a Sasakian manifold is a divergence.

The intrinsic geometry on bounded pseudoconvex domains

The Diederich--Fornæss index has been introduced since 1977 to classify bounded pseudoconvex domains. In this article, we derive several intrinsic, geometric conditions on boundary of domains for arbitrary indexes. Many results, in the past, by various mathematicians estimated the index by assuming some properties of domains. Our motivation of this paper is, the other way around, to look for how the index effects properties and shapes of domains. Especially, we look for a necessary condition of all bounded pseudoconvex domains $Ω\subset\mathbb{C}^2$ with the Diederich--Fornæss index 1. We also show that, when the Levi-flat set of $\partialΩ$ is a closed Riemann surface, then the necessary condition can be simplified.

Derivatives at the Boundary for Analytic Lipschitz Functions

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $α$ , with $0<α<1$ , in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We show that whenever such a bounded point derivation exists at a boundary point $b$ , it may be evaluated by taking a limit of classical difference quotients, for approach from a set having full area density at $b$ .

preprint2012arXiv

Holomorphic mappings between pseudoellipsoids in different dimensions

We give a necessary and sufficient condition for the existence of nondegenerate holomorphic mappings between pseudoellipsoidal real hypersurfaces, and provide an explicit parametrization for the collection of all such mappings (in the situations where they exist).

preprint2004arXiv

Existence of holomorphic sections and perturbation of positive line bundles over $q$ --concave manifolds

By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on the manifold. The bound is given explicitely in terms of the volume of the domain in this metric and a certain integral on the boundary involving the defining function and its Levi form. As application we study the perturbattion of the complex structure of a q-concave manifold.

preprint2010arXiv

CR transversality of holomorphic mappings between generic submanifolds in complex spaces

We show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Linda P. Rothschild and the first author in a paper from 2006.

On the embeddability of real hypersurfaces into hyperquadrics

In this paper, we provide {\em effective} results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, for any $N >n \geq 1$ , the defining functions $φ(z,\bar z,u)$ of all real-analytic hypersurfaces $M=\{v=φ(z,\bar z,u)\}\subset\mathbb C^{n+1}$ containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric $\mathcal Q\subset\mathbb C^{N+1}$ satisfy an {\em universal} algebraic partial differential equation $D(φ)=0$ , where the algebraic-differential operator $D=D(n,N)$ depends on $n, N$ only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every $n,N$ as above there exists $μ=μ(n,N)$ such that a Zariski generic real-analytic hypersurface $M\subset\mathbb C^{n+1}$ of degree $\geq μ$ is not transversally holomorphically embeddable into any hyperquadric $\mathcal Q\subset\mathbb C^{N+1}$ . We also provide an explicit upper bound for $μ$ in terms of $n,N$ . To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a

Certain properties of a new subclass of analytic and p-valently close-to-convex functions

In the present paper we introduce and investigate an interesting subclass K_{s}^{(k)}(γ,p) of analytic and p-valently close-to-convex functions in the open unit disk U. For functions belonging to this class, we derive several properties as the inclusion relationships and distortion theorems. The various results presented here would generalize many known recent results.

Local monomialization of a system of first integrals of Darboux type

Given a real- or complex-analytic singular foliation $θ$ with $n$ first integrals of meromorphic or Darboux type $(f_1,\dots,f_n)$ , we prove that there exists a local monomialization of the first integrals. In particular, if $θ$ is generated by the $n$ first integrals, we prove the existence of a local reduction of singularities of $θ$ to monomial singularities.

Classification of Kaehler homogeneous manifolds of non-compact dimension two

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup such that $X := G/H$ is Kaehler and the codimension of the top non-vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or equal to two. We show that $X$ is biholomorphic to a complex homogeneous manifold constructed using well-known basic building blocks, i.e., $\mathbb C, \mathbb C^*$ , Cousin groups, and flag manifolds.

Horoballs and iteration of holomorphic maps on bounded symmetric domains

Given a fixed-point free compact holomorphic self-map $f$ on a bounded symmetric domain $D$ , which may be infinite dimensional, we establish the existence of a family $\{H(ξ, λ)\}_{λ>0}$ of convex $f$ -invariant domains at a point $ξ$ in the boundary $\partial D$ of $D$ , which generalises completely Wolff's theorem for the open unit disc in $\mathbb{C}$ . Further, we construct horoballs at $ξ$ and show that they are exactly the $f$ -invariant domains when $D$ is of finite rank. Consequently, we show in the latter case that the limit functions of the iterates $(f^n)$ with weakly closed range all accumulate in one single boundary component of $\partial D$ .

Localization for Uniform Algebras Generated by Real-Analytic Functions

It is shown that if $A$ is a uniform algebra generated by real-analytic functions on a suitable compact subset $K$ of a real-analytic variety such that the maximal ideal space of $A$ is $K$ , and every continuous function on $K$ is locally a uniform limit of functions in $A$ , then $A=C(K)$ . This gives an affirmative answer to a special case of a question from the Proceedings of the Symposium on Function Algebras held at Tulane University in 1965.

Uniform approximation of Bloch functions and the boundedness of the integration operator on $H^\infty$

We obtain a necessary and sufficient condition for the operator of integration to be bounded on $H^\infty$ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions. This gives a full characterization of symbols of certain Volterra operators that act on bounded analytic functions in the disc if the symbol is assumed to be univalent. Without this assumption the answer is not known, and as the example at the end of the paper shows, the natural answer is definitely false.

Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\|pf-1\|_α$ for a given function $f$ . For $α\in [0,1]$ (which includes the Hardy and Dirichlet spaces of the disk) and general $f$ , we show that such extremal polynomials are non-vanishing in the closed unit disk. For negative $α$ , the weighted Bergman space case, the extremal polynomials are non-vanishing on a disk of strictly smaller radius, and zeros can move inside the unit disk. We also explain how $\mathrm{dist}_{D_α}(1,f\cdot \mathcal{P}_n)$ , where $\mathcal{P}_n$ is the space of polynomials of degree at most $n$ , can be expressed in terms of quantities associated with orthogonal polynomials and kernels, and we discuss methods for computing the quantities in question.

A Central Limit Theorem for Fluctuations in Polyanalytic Ginibre Ensembles

We study fluctuations of linear statistics in Polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our main result is asymptotic normality of fluctuations, extending a result of Rider and Virág. As in the analytic case, the variance is composed of independent terms from the bulk and the boundary. Our methods rely on a structural formula for polyanalytic polynomial Bergman kernels which separates out the different pure $q$ -analytic kernels corresponding to different Landau levels. The fluctuations with respect to these pure $q$ -analytic Ginibre ensembles are also studied, and a central limit theorem is proved. The results suggest a stabilizing effect on the variance when the different Landau levels are combined together.

An equivariant parametric Oka principle for bundles of homogeneous spaces

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein space $X$ , such that the fibre of $E$ is a homogeneous space of the fibre of $\mathscr G$ , with the complexification $K^\mathbb C$ of a compact real Lie group $K$ acting on $X$ , $\mathscr G$ , and $E$ . Our main result is that the inclusion of the space of $K^\mathbb C$ -equivariant holomorphic sections of $E$ over $X$ into the space of $K$ -equivariant continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch's classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.

Quaternionic Wiener Algebras, Factorization and Applications

We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem for the quaternionic matrix-valued Wiener algebras (discrete and continuous) and explore the connection to the Riemann-Hilbert problem in that setting. As applications, we characterize solvability of two classes of quaternionic functional equations and give an explicit formula for the canonical factorization of quaternionic rational matrix functions via realization.

The Kohn-Laplace equation on abstract CR manifolds: Global regularity

Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$ -dimensional, abstract CR manifold of hypersurface type, $n\geq 2$ . We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$ -forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$ . In the case that $q_0=1$ , we also establish continuity for $G_0$ and $G_n$ . Additionally, the $\bar\partial_b$ -equation on $M$ can be solved in $C^\infty(M)$ . (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$ -forms, then $G_q$ is a continuous operator on $H^s_{0,q}(M)$ and is therefore globally regular on $M$ for degrees $q_0\le q\le n-q_0$ ; and also for the top degrees $q=0$ and $q=n$ in the case $q_0=1$ . We also introduce the notion of a "plurisubharmonic CR manifold" and show that it generalizes the notion of "plurisubharmonic defining function" for a a domain in $\mathbb C^N$ and implies that $M$ satisfies the weak compactness property.

Littlewood-Paley formulas and Carleson measures for weighted Fock spaces induced by $A_\infty$ -type weights

We obtain Littlewood-Paley formulas for Fock spaces $\mathcal{F}^q_{β,ω}$ induced by weights $ω\in A^{restricted}_\infty=\cup_{1\le p<\infty}A^{restricted}_{p}$ , where $A^{restricted}_{p}$ is the class of weights such that the Bergman projection $P_α$ , on the classical Fock space $\mathcal{F}^2_α$ , is bounded on $\mathcal{L}^p_{α,ω}:=\left\{f:\, \int_{\mathbb{C}}|f(z)|^pe^{-p\fracα{2}|z|^2}\,ω(z)dA(z)<\infty \right\}.$ Using these equivalent norms for $\mathcal{F}^q_{β,ω}$ we characterize the Carleson measures for weighted Fock-Sobolev spaces $\mathcal{F}^{q,n}_{β,ω}$ .