Theory Of Several Complex Variables Research Articles

Real Hypersurfaces, Orders of Contact, and Applications

A fundamental problem in the modem theory of several complex variables concerns the boundary behavior of the Cauchy-Riemann equations. Suppose that D is an open domain in Cn, and that its boundary, M, is a smooth real submanifold of Cn. How does the geometry of M influence the function theory on D? One approach to this question is the a-Neumann problem, [6]. There one studies the inhomogeneous Cauchy-Riemann equations au -a where a is a (0, q) form with distribution coefficients on the closed domain D U M satisfying the necessary compatibility condition a a 0. The a-Neumann problem constructs a particular solution u which is orthogonal to the null space of the complex Laplacian. Kohn [10a] has solved this problem and has shown that good local regularity properties for u in terms of a follow from so-called subelliptic estimates. It is natural to seek geometric conditions on M that are necessary and sufficient for these estimates. In this paper we will be concerned with a certain geometric condition on M that we feel is the right one for subellipticity. Although the motivation for the questions discussed here comes from partial differential equations, the techniques come from algebraic geometry. To see why, we state theorems of Kohn, Greiner, and Catlin concerning subellipticity. Kohn [1Oa] has discovered a sufficient condition for a subelliptic estimate near a fixed point p in M. In case D is pseudoconvex and is contained in C2, this condition is equivalent to the finiteness of order of tangency of every complex analytic (one dimensional) manifold with M at p. Greiner [13] established the necessity of this condition in

Weighted trigonometric approximation and inner—Outer functions on higher dimensional Euclidean spaces

In this paper we consider a class of weighted trigonometric approximation problems in ip,(~%‘~). Many of these questions go back to the classic papers of Beurling [ 1,2]. One may describe the general problem as that of finding the best approximation to an arbitrary function from the closed subspace generated by functions of the form er@‘.‘+‘)f(w), where f is a fixed function and where the arguments u are to be taken from a given set S. For 9z(9”) and S a half-line, these problems, and their solution, are intimately related to the theory of inner and outer functions. The latter theory does not exist for 9”, n > 2. However, by use of functional analytic and group theoretic methods we are able to treat the trigonometric approximation problem for 4p2(9”) with S either a half-space or a quadrant. In so doing, we also arrive at a beginning of a new theory of inner and outer functions for 9”. The theory of inner and outer functions also plays a central role in the prediction problem for one-parameter processes; see Dym and McKean [4]. In Gustafson and Misra [5] regular one-parameter processes were characterized in terms of the canonical commutation relations of quantum mechanics. Particular emphasis was placed on the use of cyclic vectors. This approach enables us to investigate the weighted trigonometric approximation problems via several-parameter regular processes without appeal to the theory of several complex variables. As a byproduct, we obtain an abstract characterization of the regular representation of 9”. This answers a question raised by Chatterji [3, p. 241. We also observe a new proof for the existence and uniqueness of outer functions on 9’. This comes out of our (new) formulation and results for inner and outer functions on 9”. The theory of inner and outer functions for compact abelian groups whose dual is linearly ordered has been developed. See, for example, Helson and

Bestimmung und Anwendung von Vekua-Resolventen

The paper is concerned with the differential equation\(V_{\bar z} = C(z,\bar z)\bar V\) with\(m^2 (\log C)_{z\bar z} + + \varepsilon C\bar C = 0,m > 0,\varepsilon = \pm 1\). The Vekua resolvents are determined by means of an associated second-order differential equation. Applications are given to pseudo-analytic functions, to a differential equation in the theory of several complex variables and to the Ernst equation in general relativity.

Some applications of Synge’s formula to the theory of several complex variables

In [10] and [11], the second author proved the following theorem by using Synge’s formula:THEOREM I. Let M be a kähler manifold with positive holomorphic bi-sectional curvature. Then every pseudoconvex domain in M is a Stein manifold.

Linear forms in algebraic points of Abelian functions. II

In this paper we continue to develop the apparatus needed for the proof of the theorem announced in (11). We retain the notation of (11) together with the assumptions made there about the field of Abelian functions. This section deals with properties of more general functions holomorphic on Cn. When n = 1 the extrapolation procedure in problems of transcendence is essentially the maximum modulus principle together with the act of dividing out zeros of an analytic function. For n > 1, however, this approach is not possible, and some mild theory of several complex variables is required. This was first used in the context of transcendence by Bombieri and Lang in (2) and (12), and we now give a brief account of the basic constructions of their papers.

A note on topological degree theory for holomorphic maps

Using degree theory, an elementary topological proof is given of some well-known results in the theory of several complex variables. In particular it is shown that a compact analytic variety consists of finitely many points.

Two‐Dimensional Quarter Space Problems in One‐Speed Transport Theory

Methods derived from the theory of several complex variables are used as a means of analyzing a class of two‐dimensional transport problems in a scattering and absorbing quarter space (0 ≤ x_1, 0 ≤ x_2, −∞ ≤ x_3 ≤ ∞) described by a linear, one‐speed Boltzmann equation. Using Fourier transformation and the Bochner decomposition, the multivariable analog of the Wiener‐Hopf factorization, we find the Green's function in transform space, which solves all source problems having a solution bounded at infinity. The transform of the density asymptotically far from the corner (x_1 = x_2 = 0) is determined explicitly, while the remainder is given in terms of the solution to a pair of Fredholm equations.

Analytic Functions of Several Complex Variables.

The theory of analytic functions of several complex variables enjoyed a period of remarkable development in the middle part of the twentieth century. After initial successes by Poincare and others in the late 19th and early 20th centuries, the theory encountered obstacles that prevented it from growing quickly into an analogue of the theory for functions of one complex variable. Beginning in the 1930s, initially through the work of Oka, then H. Cartan, and continuing with the work of Grauert, Remmert, and others, new tools were introduced into the theory of several complex variables that resolved many of the open problems and fundamentally changed the landscape of the subject. These tools included a central role for sheaf theory and increased uses of topology and algebra. The book by Gunning and Rossi was the first of the modern era of the theory of several complex variables, which is distinguished by the use of these methods. The intention of Gunning and Rossi's book is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory of analytic spaces. Fundamental concepts and techniques are discussed as early as possible. The first chapter covers material suitable for a one-semester graduate course, presenting many of the central problems and techniques, often in special cases. The later chapters give more detailed expositions of sheaf theory for analytic functions and the theory of complex analytic spaces. Since its original publication, this book has become a classic resource for the modern approach to functions of several complex variables and the theory of analytic spaces.

A type of convexity in the space of 𝑛-complex variables

Introduction. This paper consists for the main part of a proof of a result in the theory of several complex variables that was obtained as an essential link in a study of the classification of homogeneous Banach algebras on compact abelian groups. (For applications of the result, see [3] and [4].) Let X be a compact subset of Cn, the space of n-complex variables. Cp(X) will denote the set of all points y in Cn that satisfy for each polynomial f

Models in the Theory of Several Complex Variables

concept has become familiar, the model is regarded as a realization of the concept. The most important disadvantage of diagrams is that they are not able to describe the manifold completely, e.g. to indicate whether a circle is to be taken with its boundary or without it. Because of this during the transition from the diagram to the abstract notion we often commit mistakes by attributing to the manifold properties which are possessed only by the model. A means of checking our conclusions would be to repeat our reasoning but now operating exclusively with abstract symbols, e.g. of the type indicated earlier in this section. This situation necessitates a certain kind of training and particular flexibility of mind, enabling one to pass from an intuitive model to an abstract formulation of notions and back. These difficulties arise naturally in operating with models of four-dimensional space; in addition to this there occur further difficulties, which we shall indicate in what follows. 4. The advantage of models in teaching and research. Despite the difficulties mentioned before it seems that in many cases the use of models for a geometrically minded individual is of great value in research. Two important reasons may perhaps be indicated: First, our interest in a thing is determined to a considerable extent by the amount of mental energy which is required for working with it. We involuntarily prefer to think in terms of familiar things which can be conceived without special effort. In other words, the extent of our interest in any subject depends to a certain degree on our previous associations-and here again we the advantage of being able to use intuitive geometry to help in reasoning about abstract relations. Second, if a relationship is known, it is not too important how we formulate it, provided only that our formulation is sufficiently precise. Research, on the other hand, is the search for new relations and in that case it becomes highly significant in which way our problems are formulated. If we can succeed in stating our problem in terms of a class of objects with which we have gained great familiarity, then we may be able to exploit our previous experience with these things. The form of the problem in the new representation may suggest comparisons and hypotheses and further questions, and our previous experience will enable us to new relationships all the more readily. Thus, the moment that complex numbers are regarded as points in a plane, the analogy with integrals of real functions leads to the formulation of curvilinear integrals in the complex plane-the most fruitful approach in the theory of functions. Furthermore, even after some fact has been discovered and it has been reduced to abstract form which completely and rigorously describes it, even then, the intuitive processes by means of which we have worked are of importance in further research because of the ready associations and the many problems which in this way arise almost spontaneously. This content downloaded from 157.55.39.180 on Tue, 27 Sep 2016 05:23:13 UTC All use subject to http://about.jstor.org/terms 1946] MODELS IN THE THEORY OF SEVERAL COMPLEX VARIABLES 499 5. Models of four-dimensional manifolds. It would, perhaps, be of interest to illustrate the foregoing by a description of a type of model used for visualization of four-dimensional space.* These models are moving three-dimensional pictures, that is to say time is chosen as one of the coordinates. A four-dimensional domain is then represented as a moving picture. The continuous process is thus recorded by a finite number of frames, and this last is in turn often replaced by a number of stills. In order to have a better idea as to how intuitive these models are, let us put ourselves in the position of beings (Flatlanders) who are able to see two-dimensional domains but who do not possess intuition for three-dimensional space. xs

Models in the Theory of Several Complex Variables

Models in the Theory of Several Complex Variables