Riemann Surfaces and the L 2 $\bar{\partial}$ -Method for Scalar-Valued Forms

The aim of this article is to extend the theory of several complex variables to the non-commutative realm. Some basic results, such as the Bochner-Martinelli formula, the existence theorem of the solutions to the non-homogeneous Cauchy-Riemann equations, and the Hartogs theorem, are generalized from complex analysis in several variables to Clifford analysis in several paravector variables. In particular, the Bochner-Martinelli formula in several paravector variables unifies the corresponding formulas in the theory of one complex variable, several complex variables, and several quaternionic variables with suitable modifications.

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.

In the classical complex analysis of several variables Lars Hoermander had been proven for Hartog's extension theorem. His proof is based on solving the inhomogeneous Cauchy–Riemann system. However, it is difficult when generalizing this diagram to prove this problem in Clifford analysis because of the non-commutative property. In this paper we will propose a method to overcome this difficultly and we will prove the Hartog's extension theorem for multi-regular function taking valued in Clifford algebra

In view of A. Andreotti and H. Grauert's vanishing theorem for q-complete domains in C^n, (Th\'eor\`eme de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193--259,) we re-prove a vanishing result by J.-P. Sha, (p-convex Riemannian manifolds, Invent. Math. 83 (1986), no. 3, 437--447,) and H. Wu, (Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), no. 3, 525--548,) for the de Rham cohomology of strictly p-convex domains in R^n in the sense of F. R. Harvey and H. B. Lawson, (The foundations of p-convexity and p-plurisubharmonicity in riemannian geometry, arXiv:1111.3895v1 [math.DG]). Our proof uses the L^2-techniques developed by L. H\"ormander, (An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library, 7, North-Holland Publishing Co., Amsterdam, 1990,) and A. Andreotti and E. Vesentini, (Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes \'Etudes Sci. Publ. Math. 25 (1965), 81--130).

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.

In this paper, we define an appropriate version of Morrey spaces for spaces of homogeneous type. As our main result, we give a sufficient condition for an operator to be bounded on the version of Morrey spaces (cf. Theorem 3.1 and Corollary 3.5). Moreover, using thin result, we prove Morrey space estimates for various operators arising in harmonic analysis and complex analysis of several variables.

A sufficient condition for global regularity of the [formula omitted]-Neumann operator

This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.

The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non-commuting variables. In this paper we introduce the class of \emph{nc-manifolds}, the mathematical objects that at each point possess a neighborhood that has the structure of an \emph{nc-domain} in the \emph{$d$-dimensional nc-universe $\m^d$}. We illustrate the use of such manifolds in free analysis through the construction of the non-commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non-commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in $\m^2$ we construct a 2-dimensional non-commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton-Girard formulae for power sums of two non-commuting variables.

Many authors, e.g., Bavrin, Jakubowski, Liczberski, Pfaltzgraff, Sitarski, Suffridge, and Stankiewicz, have discussed some families of holomorphic functions of several complex variables described by some geometrical or analytical conditions. We consider a family of holomorphic functions of several complex variables described in n-circular domain of the space C n . We investigate relations between this family and some of type of Bavrin’s families. We give estimates of G-balance of k-homogeneous polynomial, a distortion type theorem and a sufficient condition for functions belonging to this family. Furthermore, we present some examples of functions from the considered class.

This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.

1. Some problems arising in the study of value distribution of functions of two complex variables. One of the objectives of modern analysis consists in the generalization of methods of the theory of functions of one complex variable in such a way that the procedures in the revised form can be applied in other fields, in particular, in the theory of functions of several complex variables, in the theory of partial differential equations, in differential geometry, etc. In this way one can hope to obtain in time a unified theory of various chapters of analysis. The method of the kernel function is one of the tools of this kind. In particular, this method permits us to develop some chapters of the theory of analytic and meromorphic functions f(z1, * * *, zn) of the class j2(I2n), various chapters in the theory of pseudo-conformal transformations (i.e., of transformations of the domains V2n by n analytic functions of n complex variables) etc. On the other hand, it is of considerable interest to generalize other chapters of the theory of functions of one variable, at first to the case of several complex variables. In particular, the study of value distribution of entire and meromorphic functions represents a topic of great interest. Generalizing the classical results about the zeros of a polynomial, Hadamard and Borel established a connection between the value distribution of a function and its growth. A further step of basic importance has been made by Nevanlinna and Ahlfors, who showed not only that the results of Hadamard and Borel in a sharper form can be obtained by using potential-theoretical and topological methods, but found in this way important new relations, and opened a new field in the modern theory of functions. As one passes from the theory of functions of one variable to the case of two and more variables(2), the question of the value distribution becomes more complex and many directions arise which can be considered as a generalization of the above mentioned chapter of the theory of functions of one variable. However, the formulation of the problems which can be answered, and generalization of methods used in one variable, is not immediate. A function of two complex variables may vanish in a domain on a segment

The differential equations of chemical kinetics are systems of nonlinear (polynomial) differential equations, therefore their solutions cannot usually be found in symbolic form. Here we offer a method to solve classes of kinetic differential equations based on the Cauchy–Riemann equations. It turns out that the method can be used to symbolically solve some polynomial differential equations that are not necessarily kinetic, as well.

In this chapter, we consider a useful generalization of the notion of a holomorphic function, namely, that of a holomorphic section of a holomorphic line bundle. We first consider the basic properties of holomorphic line bundles as well as those of sheaves and divisors. We then proceed with a discussion of the solution of the inhomogeneous Cauchy–Riemann equation with L 2 estimates in this more general setting. In this setting, there is a natural generalization of Theorem 2.9.1 for Hermitian holomorphic line bundles (E,h) with positive curvature; that is, iΘ h >0, where Θ h is a natural generalization of the curvature form \(\Theta_{\varphi}=\partial\bar{\partial} \varphi \) considered in Sect. 2.8. In fact, Sects. 3.6–3.9 may be read in place of most of the material in Sects. 2.6–2.9. We then consider applications, mostly to the study of holomorphic line bundles on open Riemann surfaces (holomorphic line bundles on compact Riemann surfaces are considered in greater depth in Chap. 4). For example, in Sect. 3.11, we prove that every holomorphic line bundle on an open Riemann surface admits a positive-curvature Hermitian metric (this follows easily from the results of Sect. 2.14); and we then obtain a slightly more streamlined proof of (a generalization of) the Mittag-Leffler theorem (Theorem 2.15.1). In Sect. 3.12, we prove the Weierstrass theorem (Theorem 3.12.1), according to which every holomorphic line bundle on an open Riemann surface is actually holomorphically trivial.KeywordsRiemann SurfaceDifferential FormCompact Riemann SurfaceHolomorphic SectionHolomorphic Line BundleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The authors obtain a generalization of Jack-Miller-Mocanu's lemma and, using the technique of subordinations, deduce some properties of holomorphic mappings from the unit polydisc in $ℂ^n$ into $ℂ^n$.