Several Complex Variables Research Articles

Complex Analysis: A Brief Tour into Higher Dimensions

1. INTRODUCTION. Why is it that most graduate students of mathematics (and many undergraduates as well) are exposed to complex analysis in one variable, yet only a small minority of students or, for that matter, professional mathematicians ever learn even the most basic corresponding theory in several variables? Think about it another way: Could anyone seriously argue that it might be sufficient to train a mathematics major in calculus of functions of one real variable without expecting him or her to learn at least something about partial derivatives, multiple integrals, and some higher dimensional version of the Fundamental Theorem of Calculus? Of course not, the real world is not one-dimensional! But neither is the complex world: witness classical applications of complex analysis to quantum field theory, more recent uses in twistor and gravitation theory, and the latest developments in string theory! Multidimensional complex analysis is an indispensable tool in modern theoretical physics. (See, for example, Green, Schwarz, and Witten [6], Manin [12], Henkin and Novikov [9].) Aside from questions of applicability, shouldn’t the pure mathematician’s mind wonder about the restriction to functions of only one complex variable? It should not surprise anyone that there is a natural extension of complex analysis to the multivariable setting. What is surprising is the many new and intriguing phenomena that appear when one considers more than one variable. Indeed, these phenomena presented major challenges to any straightforward generalization of familiar theorems. Amazing progress was made in the 1940s and 1950s, when the area was enriched by new and powerful tools such as coherent analytic sheaves and their cohomology theory. Yet for decades these apparently quite abstract techniques constituted a formidable barrier that deterred analysts from exploring this territory unless they were committed to a research career in multidimensional complex analysis. Fortunately, many of the technical hardships of the pioneering days can now be overcome with much less effort by approaching the subject along different routes. The purpose of this article is to lead you, as painlessly as possible, on a tour through some of the foundations of complex analysis in several variables, and to take you to some vantage points from which we can enjoy some of the fascinating sights that remain hidden from us as long as we restrict ourselves to the complex plane. Along the way, we shall encounter a few of the unexpected higher dimensional phenomena and explore fundamental new concepts, always trying to gain an understanding of the underlying difficulties. Good walking shoes (i.e., an elementary introduction to basic complex analysis in one variable, and standard multivariable real calculus) are all the equipment that is needed, no technical gear or skills will be required. I will be rewarded if, at the end of our hike, many of you will recommend this tour to friends and colleagues. Perhaps a few of you will be sufficiently intrigued and challenged that you will pick up one of the excellent books that are available to guide you through more advanced terrain in order to reach the “high peaks” of the subject.

On perturbations of pluriregular sets generated by sequences of polynomial maps

It is shown that an infinite sequence of polynomial mappings of several complex variables, with suitable growth restrictions, determines a filled-in Julia set which is pluriregular. Such sets depend continuously and analytically on the generating sequence

Inequalities from complex analysis

Inequalities from Complex Analysis is a careful, friendly exposition of inequalities and positivity conditions for various mathematical objects arising in complex analysis. The author begins by defining the complex number field, and then discusses enough mathematical analysis to reach recently published research on positivity conditions for functions of several complex variables. The development culminates in complete proofs of a stabilization theorem relating two natural positivity conditions for real-valued polynomials of several complex variables. The reader will also encounter the Bergman kernel function, Fourier series, Hermitian linear algebra, the spectral theorem for compact Hermitian operators, plurisubharmonic functions, and some delightful inequalities. Numerous examples, exercises, and discussions of geometric reasoning appear along the way. Undergraduate mathematics majors who have seen elementary real analysis can easily read the first five chapters of this book, and second year graduate students in mathematics can read the entire text. Some physicists and engineers may also find the topics and discussions useful. The inequalities and positivity conditions herein form the foundation for a small but beautiful part of complex analysis. John P. D'Angelo was the 1999 winner of the Bergman Prize; he was cited for several important contributions to complex analysis, including his work on degenerate Levi forms and points of finite type, as well as work, some joint with David Catlin, on positivity conditions in complex analysis

Norm preserving extensions of holomorphic functions from subvarieties of the bidisk

A basic result in the theory of holomorphic functions of several complex variables is the following special case of the work of H. Cartan on the sheaf cohomology on Stein domains ([10], or see [14] or [16] for more modern treatments).

Kernel convergence and biholomorphic mappings in several complexvariables

We deal with kernel convergence of domains in ℂn which are biholomorphically equivalent to the unit ball B. We also prove that there is an equivalence between the convergence on compact sets of biholomorphic mappings on B, which satisfy a growth theorem, and the kernel convergence. Moreover, we obtain certain consequences of this equivalence in the study of Loewner chains and of starlike and convex mappings on B.

A study on bornological properties of the space of entire functions of several complex variables

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of
 topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$
 and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

Some properties of Reinhardt and Hartogs domains

The domains of certain types, such as Reinhardt and Hartogs, are used in different problems of theory of functions of several complex variables. For instance, any power series of several complex variables converges in the complete logarithmically convex Reinhardt domain. Transformations of Reinhardt and Hartogs allow us to diminish the dimension of space and to consider some type of domains using its images. This allows the visibility of the geometric representation and the simplification of the study of properties of such domains. In this article we consider some properties of Reinhardt and Hartogs domains and transformations. The properties of domain conservation under Reinhardt transformation and convexity conservation under Reinhardt and Hartogs transformation are proved.

Uniqueness problem of meromorphic mappings in several complex variables for moving targets

Nevanlinna showed that for two nonconstant meromorphic functions on the complex plane, if they have the same inverse images counting multiplicities for four distinct values, then they coincide up to a Möbius transformation, and if they have the same inverse images for five distinct values, then they coincide. Fujimoto and Smiley extended Nevanlinna's uniqueness theorems to the case of meromorphic mappings of several complex variables into the complex projective space for hyperplanes. Recently, Motivated by Ru Min and Stoll's accomplishment of the second main theorem for moving targets, Li Baoqin and Shirosaki proved some uniqueness theorems of entire functions in several complex variables and meromorphic functions in one complex variable, respectively, for moving targets. Using the techniques of value distribution theory in several complex variables, we prove some uniqueness theorems of meromorphic mappings of several complex variables into the complex projective space for moving targets.

Best ${L^p}$-approximation of generalized biaxially symmetric potentials over Carath'eodory domains in ${C^N}$ having slow growth

Let $ F$ be a real valued generalized biaxially symmetric potentials (GBASP) defined on the Caratheodory domain on $ C^N$. Let $ L_\mu^p(D)$ be the class of all functions $ F$ holomorphic on $ D$ such that $ \parallel F\parallel_{D,p}=[\int_D\mid F\mid^pd\mu]^{1\over p}$. Where $ \mu$ is the positive finite, Boral measure with regular asymptotic distribution on $ C^N$. For $ F\in L_{\mu}^p(D)$, set $ E_n^p(F)=\inf\{\parallel F-P\parallel_{D,p}:P\in H_n\}$, $ H_n$ consist of all real biaxisymmetric harmonic polynomials of degree at most $ 2n$. The paper deals with the growth of entire function GBASP in terms of approximation error in $ L_{\mu}^p$-norm on $ D$. The analysis utilizes the Bergman and Gilbert integral operator method to extend results from classical function theory on the best polynomial approximation of analytic functions of several complex variables. Finally we prove a generalized decomposition theorem in a new way. The paper is the generalization of the concepts of generalized growth parameters to entire functions on Caratheodory domains on $ C^N$ (instead of entire holomorphic functions on $ C$) for slow growth.

The uniqueness question in the multidimensional moment problem with applications to phase space observables

The theory of holomorphic functions of several complex variables is applied in proving a multidimensional variant of a theorem involving an exponential boundedness criterion for the classical moment problem. A theorem of Petersen concerning the relation between the multidimensional and one-dimensional moment problems is extended for half-lines and compact subsets of the real line. These results are used to solve the moment problem for the quantum phase space observables generated by the number states.

Parametric Representation of Univalent Mappings in Several Complex Variables

AbstractLet B be the unit ball of with respect to an arbitrary norm. We prove that the analog of the Carathéodory set, i.e. the set of normalized holomorphic mappings from B into of “positive real part”, is compact. This leads to improvements in the existence theorems for the Loewner differential equation in several complex variables. We investigate a subset of the normalized biholomorphic mappings of B which arises in the study of the Loewner equation, namely the set S0(B) of mappings which have parametric representation. For the case of the unit polydisc these mappings were studied by Poreda, and on the Euclidean unit ball they were studied by Kohr. As in Kohr’s work, we consider subsets of S0(B) obtained by placing restrictions on the mapping from the Carathéodory set which occurs in the Loewner equation. We obtain growth and covering theorems for these subsets of S0(B) as well as coefficient estimates, and consider various examples. Also we shall see that in higher dimensions there exist mappings in S(B) which can be imbedded in Loewner chains, but which do not have parametric representation.

On Meromorphically Normal Families of Meromorphic Mappings of Several Complex Variables into PN(C)

We shall prove some meromorphic normality criteria for families of meromorphic mappings of several complex variables into PN(C), the complex N-dimensional projective space, related to Nochka's Picard type theorems. Some related results (e.g., extension theorems, improved normality criteria, and quasi-normality criteria) will be obtained also. The technique in this paper mainly depends on Stoll's normality criteria for families of non-negative divisors on a domain of Cn.

On the Gap in the Spectrum of the Translations

The spectrum of the translations in local quantum field theory will be analyzed in order to show that in a positive energy representation without vacuum vector and with lowest mass m 1 there is no gap in the spectrum which is larger than 2m 1. In particular in a zero mass representation there is no hole at all. These results are obtained with methods of analytic functions of several complex variables.

A NOTE ON THURSTON-WINKELNKEMPER'S CONSTRUCTION OF CONTACT FORMS ON 3-MANIFOLDS

A contact structure on a closed oriented 3-manifold 3 is a completely nonintegrable plane field on it. The existence of a contact structure on 3 was first proved by Martinet [14] and then Lutz [12], [13] for each homotopy class of plane fields. A co-orientable contact structure is given as the kernel of a 1-form α with α ∧ α > 0 or < 0, which is called a positive or negative contact form respectively. Thurston and Winkelnkemper [17] deduced the existence of a contact form in an elegant way from Alexander’s theorem [1] on open-book decompositions. Eliashberg [2] showed that the most of these contact structures are, however, too flexible for geometrical interest (see §2). On the other hand, he completely characterized more historic examples related closely to the complex analysis in several variables, namely, the strictly pseudo-convex boundary of compact Stein surfaces ([4]). The symplectic fillability is one of its significant generalizations. Recently Loi and Piergallini [11] translated Eliashberg’s characterization of compact Stein surfaces into the language of Lefschetz fibration by using Gompf’s method [8]. They also showed that 3 is realizable as the boundary of a Stein surface if and only if it admits an open-book decomposition whose monodromy map is a composition of right-handed Dehn-twists. In this note, we improve Thurston-Winkelnkemper’s construction so that we obtain a symplectically fillable contact structure in the case where the monodromy map of a given open-book decomposition is a composition of right-handed Dehn-twists along mutually disjoint curves (Theorem 2). An interesting by-product of our construction is Theorem 3 in §4 which gives a deformation of symplectically fillable contact structures into a foliation with a Reeb component. Note that a foliation admitting a Reeb component itself is not symplectically fillable. I would like to thank the referee for noticing the importance of Theorem 3.

An Extension Theorem and Subclasses of Univalent Mappings in Several Complex Variables

Let B be the unit ball in C n and let U be the unit disc in C. The aim of this work is to construct a family of operators Ψ n,α that provide a way to extend a locally univalent function ƒ ∈ H(U) to a locally univalent mapping Fα ∈ H(B), where α ∈ (0,1]. If ƒ is normalized univalent, then Fα can be imbedded in a Loewner chain. Also if ƒ ∈ S*, then Fα is starlike. We show that if ƒ belongs to a class of univalent functions which satisfy growth and distortion results, then the mapping Fα satisfies similar growth and distortion results. Also we study the concept of linear-invariant families as it relates to families generated by the operator Ψ n,0, and we obtain in this way another example of a L.I.F. that has minimum order (n + 1)/2 and is not a subset of the normalized convex mappings in the unit ball of C n (for n ≤ 2.)

Rigidity Theorems for Multiparametric Deformations of the Universal Enveloping Algebras

The well-known one-dimensional Riemann–Hilbert problem on C concerns the proof of the existence of the Fuchsian system on C with a given monodromy, that is, with a given representation of a fundamental group, which is a braid group. In higher dimensions, the classical statement of the problem is modified so that instead of proving the existence of a system of differential equations of Fuchsian type, the problem of characterization of the representations of a fundamental group that can be a monodromy representation of the differential system of Fuchsian type is stated. It is a purely algebraic problem, but since it is closely connected with the question concerning the recovery of the functions of several complex variables if their ramification is given, its solution is based on analytical methods connected with properties of the power series on the parameters of the differential equation. In the context of the Riemann–Hilbert problem, to these methods should be added some algebraic ones concerning the algebraic structures that can be defined on algebras of such series. Some of these methods are considered in this paper, in particular, the method of multi-parametric deformations of some algebras, more exactly, the quantum enveloping algebras. In the one-parametric case, the quantum enveloping algebras are the algebras over the ring of the formal power series C[[h]]. The last object carries the topological structure defined by the powers of an ideal generated by a (quantum) parameter h. This topological structure naturally induces some structures on algebras and on modules over algebras. All methods of proving the different statements about these structures naturally are based on induction on the powers of h. Obviously, such a technique can be extended to the formal power series of several variables C[[h1, . . . , hn]]. Since many of the proofs of the theorems are easily carried over from the one-parametric case to the multi-parametric one, we have hopes that a variety of facts from the classical theory of quantum groups are valid for multi-parametric deformations. This paper is devoted to problems connected with multi-parametric generalizations of methods that are important for the multi-parametric analogue of the Drinfeld–Kohno theory. The well-known Drinfeld–Kohno theorem is a statement about the conditions of solvability of the Riemann–Hilbert problem in a class of the Knizhnik–Zamolodchikov equations [2, 7], which are particular cases of the multidimensional Fuchsian systems. Their proof and some of their generalizations can be found in [2,3,5,7]. In this paper, we first introduce some definitions that are used in the extensions of the Drinfeld– Kohno theory: the definition of the multi-parametric Drinfeld–Jimbo algebra and definitions of the braided multi-parametric quantum enveloping algebra and the braided multi-parametric quasi-bialgebra. We introduce the isomorphism of the quasi-bialgebras up to the twisting. Further, the uniqueness theorems for the homomorphisms of the trivial multi-parametric deformation of the enveloping algebra if these homomorphisms coincide on the universal enveloping algebras are proved. Then we prove some theorems on the uniqueness for the quantum enveloping algebras and for the Drinfeld–Jimbo algebra. We also prove the theorem on the isomorphism (up to the twisting) of the braided quasi-bialgebras having different associators. All these results are the necessary components in a proof of the multi-parametric analogue of the Drinfeld theorem on the isomorphism of the structure of a braided quasi-bialgebra (on

Interpolation with estimates in C{sup n} and its applications

The problem of interpolation in spaces of entire functions of several complex variables of finite order and type is studied. The extension is performed from the zero set of an entire function f of finite order and type. Sufficient conditions for interpolation are obtained in terms of lower bounds for the first derivatives of f that are non-zero at the points in its zero set. The result is applied to the problem of the validity of the fundamental principle in spaces of solutions of homogeneous convolution equations.

Interpolation with estimates in and its applications

The problem of interpolation in spaces of entire functions of several complex variables of finite order and type is studied. The extension is performed from the zero set of an entire function f of finite order and type. Sufficient conditions for interpolation are obtained in terms of lower bounds for the first derivatives of f that are non-zero at the points in its zero set. The result is applied to the problem of the validity of the fundamental principle in spaces of solutions of homogeneous convolution equations.

Application of ϵ-entropy theory to Kolmogorov—Arnold representation theorem

It is shown that Kolmogorov—Arnold representation theorem, which is the solution to the 13th problem formulated by D. Hilbert and holds in the theory of real-valued continuous functions of several real variables, does not hold in the theory of complex-valued analytic functions of several complex variables.

Condition for global existence of holomorphic solutions of a certain differential equation on a Stein domain of $C^{n+1}$ and its applications

We give a necessary and sufficient condition for the existence of global solutions of some partial differential equation which is locally solvable and give some applications in complex analysis of several variables.