Domain Of Holomorphy Research Articles

A New Characterization of $$L^2$$-Domains of Holomorphy with Null Thin Complements via $$L^2$$-Optimal Conditions

A New Characterization of $$L^2$$-Domains of Holomorphy with Null Thin Complements via $$L^2$$-Optimal Conditions

Properties of Bounded Holomorphic Functions: A Survey

We discuss interrelations between {mathcal {H}}^infty -convex domains and {mathcal {H}}^infty -domains of holomorphy for various classes of domains in {mathbb {C}}^n.

A holomorphic mapping property of analytic pseudo-differential operators

We study the holomorphic extendibility of $\text{Op}(p)u$, when $p$ is an analytic symbol, and explicit information is available on the domains of holomorphic extendibility of both $p$ and $u$. By a contour deformation argument, we obtain a precise local estimate of the domain of holomorphy of $\text{Op}(p)u$ in terms of the information on $p$ and $u$.

Non-negative divisors and the Grauert metric

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics on the complement of a principal divisor in \(\mathbb {C}^n\), \(n \ge 1\). In addition, we also study how this metric and its holomorphic sectional curvature behave when the corresponding principal divisors vary continuously.

Properties of Bounded Holomorphic Functions: A Survey

We discuss interrelations between H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^\\infty $$\\end{document}-convex domains and H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^\\infty $$\\end{document}-domains of holomorphy for various classes of domains in Cn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}^{n}$$\\end{document}.

On Grauert’s examples of complete Kähler metrics

Grauert showed that the existence of a complete Kähler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, C n ∖ { 0 } , n ≥ 2 \mathbb {C}^n \setminus \{0\}, n \ge 2 and B N ∖ A \mathbb {B}^N \setminus A , N ≥ 2 N \ge 2 and A ⊂ B N A \subset \mathbb {B}^N is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of the restriction of these metrics to the leaves of a suitable holomorphic foliation in these two examples. We also examine this metric on the punctured plane C ∗ \mathbb {C}^{\ast } and show that it behaves very differently in this case.

Topological convexity in complex surfaces

We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out open domains of holomorphy (Stein surfaces), but they are much more common. We provide tools for constructing TPC embeddings, and show that every closed, oriented 3-manifold M has a TPC embedding in a compact, complex surface (without boundary) realizing any homotopy class of almost-complex structures (the analogue of the homotopy class of the contact plane field in the smooth case). We prove our tool theorems with invariants that classify almost-complex structures on any 4-manifold homotopy equivalent to M. These invariants are amenable to computation and respected by homeomorphisms (not necessarily smooth). We study the two equivalence classes of smoothings on the product of a 3-manifold with a line, and on collared ends. Both classes of smoothings are realized by holomorphic embeddings exhibiting any preassigned homotopy class of almost-complex structures. One class arises from TPC embedded 3-manifolds, while the other likely does not.

Self-consistent O(4) model spectral functions from analytically continued functional renormalization group flows

In this paper we explore practicable ways for self-consistent calculations of spectral functions from analytically continued functional renormalization group (aFRG) flow equations. As a particularly straightforward one we propose to include parametrizations of self-energies based on explicit analytic one-loop expressions. To exemplify this scheme we calculate the spectral functions of pion and sigma meson of the $O(4)$ model at vanishing temperature in the broken phase. Comparing the results with those from previous aFRG calculations, we explicitly demonstrate how self-consistency at all momenta fixes the tight relation between particle masses and decay thresholds. In addition, the two-point functions from our new semianalytic FRG scheme have the desired domain of holomorphy built in and can readily be studied in the entire cut-complex frequency plane, on physical as well as other Riemann sheets. This is very illustrative and allows, for example, to trace the flow of the resonance pole of the sigma meson across an unphysical sheet. In order to assess the limitations due to the underlying one-loop structure, we also introduce a fully self-consistent numerical scheme based on spectral representations with scale-dependent spectral functions. The most notable improvement of this numerically involved calculation is that it describes the three-particle resonance decay of an off-shell pion, ${\ensuremath{\pi}}^{*}\ensuremath{\rightarrow}\ensuremath{\sigma}\ensuremath{\pi}\ensuremath{\rightarrow}3\ensuremath{\pi}$. Apart from this further conceptual improvement, overall agreement with the results from the considerably simpler semianalytic one-loop scheme is very encouraging, however. The latter can therefore provide a sound and practicable basis for self-consistent calculations of spectral functions in more realistic effective theories for warm and dense matter.

Exponential ReLU DNN Expression of Holomorphic Maps in High Dimension

For a parameter dimension din {mathbb {N}}, we consider the approximation of many-parametric maps u: [-,1,1]^drightarrow {mathbb R} by deep ReLU neural networks. The input dimension d may possibly be large, and we assume quantitative control of the domain of holomorphy of u: i.e., u admits a holomorphic extension to a Bernstein polyellipse {{mathcal {E}}}_{rho _1}times cdots times {{mathcal {E}}}_{rho _d} subset {mathbb {C}}^d of semiaxis sums rho _i>1 containing [-,1,1]^{d}. We establish the exponential rate O(exp (-,bN^{1/(d+1)})) of expressive power in terms of the total NN size N and of the input dimension d of the ReLU NN in W^{1,infty }([-,1,1]^d). The constant b>0 depends on (rho _j)_{j=1}^d which characterizes the coordinate-wise sizes of the Bernstein-ellipses for u. We also prove exponential convergence in stronger norms for the approximation by DNNs with more regular, so-called “rectified power unit” activations. Finally, we extend DNN expression rate bounds also to two classes of non-holomorphic functions, in particular to d-variate, Gevrey-regular functions, and, by composition, to certain multivariate probability distribution functions with Lipschitz marginals.

Increasing unions of Stein spaces with singularities

We show that if $X$ is a Stein space and, if $\Omega\subset X$ is exhaustable by a sequence $\Omega_{1}\subset\Omega_{2}\subset\ldots\subset\Omega_{n}\subset\dots$ of open Stein subsets of $X$, then $\Omega$ is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for $X=\mathbb{C}^{n}$ and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When $X$ has dimension $2$, we prove that the same result follows if we assume only that $\Omega\subset\subset X$ is a domain of holomorphy in a Stein normal space. It is known, however, that if $X$ is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets $X_{1}\subset X_{2}\subset\dots\subset X_{n}\subset\dots$, it does not follow in general that $X$ is holomorphically-convex or~holomorphically-separate (even if $X$ has no singularities). One can even obtain $2$-dimensional complex manifolds on which all holomorphic functions are constant.

Representations of closed quadratic forms associated with Stieltjes and inverse Stieltjes holomorphic families of linear relations

In this paper holomorphic families of linear relations that belong to the Stieltjes or inverse Stieltjes class are studied. It is shown that in their domain of holomorphy $\mathbb{C}\setminus\mathbb{R}_+$ the values of Stieltjes and inverse Stieltjes families are, up to a rotation, maximal sectorial. This leads to a study of the associated closed sesquilinear forms and their representations. In particular, it is shown that the Stieltjes and inverse Stieltjes holomorphic families of linear relations are of type (B) in the sense of Kato. These results are proved by using linear fractional transforms which connect these families to holomorphic functions that belong to a combined Nevanlinna-Schur class and a key tool then relies on a specific structure of contractive operators. <br><br> Розглядаються голоморфні сім’ї лінійних відношень, які належать до класу Стілтьєса та оберненого класу Стілтьєса. Показано, що в їхній області голоморфності $\mathbb{C}\setminus\mathbb{R}_+$ значення цих сімей є, з точністю до обертання, максимальними секторіальними. Із цим пов’язане дослідження відповідних замкнених півторалінійних форм та їхніх представлень. Зокрема, показано, що стілтьєсівські та обернені стілтьєсівські голоморфні сім’ї лінійних відношень належать до типу (В) у сенсі Като. Доведення базується на використанні дробово-лінійних перетворень, які переводять розглядувані сім’ї в голоморфні функції класу Неванлінни-Шура, псля чого використовується спеціальні структури операторів стиску.

Analyticity and causality in conformal field theory

We investigate analyticity properties of correlation functions in conformal field theories (CFTs) in the Wightman formulation. The goal is to determine domain of holomorphy of permuted Wightman functions. We focus on crossing property of three-point functions. The domain of holomorphy of a pair of three-point functions is determined by appealing to Jost’s theorem and by adopting the technique of analytic completion. This program paves the way to address the issue of crossing for the four-point functions on a rigorous footing.

An interpolation property of locally Stein sets

We prove that, if D is a normal open subset of a Stein space X of puredimension such that D is locally Stein at every point of ∂D n Xsg, then, for every holomorphic vector bundle E over D and every discrete subset Ʌ of D Xsg whose set of accumulation points lies in ∂D Xsg, there is a holomorphic section of E over D with prescribed values on Ʌ. We apply this to the local Steinness problem and domains of holomorphy.

From universality to generic non-extendability and total unboundedness in spaces of holomorphic functions

It is well known that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. If $$X({\varOmega })$$ is a space of holomorphic functions we extend these notions to $$X({\varOmega })$$ -domain of holomorphy and weak $$X({\varOmega })$$ -domain of holomorphy. For several function spaces $$X(\varOmega )$$ , satisfying weak assumptions, we prove that the notions of $$X(\varOmega )$$ -domain of holomorphy and weak $$X(\varOmega )$$ -domain of holomorphy are equivalent and that in this case the set of non-extendable functions in $$X({\varOmega })$$ is a dense $$G_\delta $$ -subset of $$X({\varOmega })$$ . Similar results are obtained for the stronger notion of total unboundedness. Finally we provide examples of new spaces $$X({\varOmega })$$ , where all the above hold. Mainly they are localized versions of classical function spaces and combinations of them.

Infinite Dimensional Holomorphic Homogeneous Regular Domains

We extend the concept of a finite dimensional {\it holomorphic homogeneous regular} (HHR) domain and some of its properties to the infinite dimensional setting. In particular, we show that infinite dimensional HHR domains are domains of holomorphy and determine completely the class of infinite dimensional bounded symmetric domains which are HHR. We compute the greatest lower bound of the squeezing function of all HHR bounded symmetric domains, including the two exceptional domains. We also show that uniformly elliptic domains in Hilbert spaces are HHR.

Extendability and domains of holomorphy in infinite-dimensional spaces

Extendability and domains of holomorphy in infinite-dimensional spaces

Topological barriers for locally homeomorphic quasiregular mappings in 3-space

We construct a new type of locally homeomorphic quasiregular mappings in the 3-sphere and discuss their relation to the M.A.Lavrentiev problem, the Zorich map with an essential singularity at infinity, the Fatou's problem and a quasiregular analogue of domains of holomorphy in complex analysis. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms $M$ with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such locally homeomorphic quasiregular mappings are defined in the 3-sphere $S^3$ as mappings equivariant with the standard conformal action of uniform hyperbolic 3-lattices $\Gamma$ in the unit 3-ball and its complement in $S^3$ and with its discrete representation $G=\rho(\Gamma)$ in the group of isometries of $H^4 $. Here $G$ is the fundamental group of our non-trivial hyperbolic 4-cobordism $M=(H^4\cup\Omega(G))/G$ and the kernel of the homomorphism $\rho \!:\! \Gamma\rightarrow G$ is a free group $F_3$ on three generators.

On B-type Open–Closed Landau–Ginzburg Theories Defined on Calabi–Yau Stein Manifolds

We consider the bulk algebra and topological D-brane category arising from the differential model of the open-closed B-type topological Landau-Ginzburg theory defined by a pair $(X,W)$, where $X$ is a non-compact Calabi-Yau manifold and $W$ has compact critical set. When $X$ is a Stein manifold (but not restricted to be a domain of holomorphy), we extract equivalent descriptions of the bulk algebra and of the category of topological D-branes which are constructed using only the analytic space associated to $X$. In particular, we show that the D-brane category is described by projective matrix factorizations defined over the ring of holomorphic functions of $X$. We also discuss simplifications of the analytic models which arise when $X$ is holomorphically parallelizable and illustrate these analytic models in a few classes of examples.

DOMAINS OF HOLOMORPHY FOR IRREDUCIBLE ADMISSIBLE UNIFORMLY BOUNDED BANACH REPRESENTATIONS OF SIMPLE LIE GROUPS

In this note, we address a question raised by B. Krotz on the classification of G-invariant domains of holomorphy for irreducible admissible Banach representations of connected non-compact simple real linear Lie groups G. When G is not of Hermitian type, we give a complete description of such G-invariant domains for irreducible admissible uniformly bounded representations on reflexive Banach spaces and, in particular, for all irreducible uniformly bounded Hilbert representations. When the group G is Hermitian, we determine such G-invariant domains only when the representations considered are highest or lowest weight representations.

Domains of holomorphy

We give a simple proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire’s Category Theorey and Montel’s Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of functions \(X=X({\varOmega })\subset H({\varOmega })\). We show that the set of non-extendable functions not only contains a \(G_{\delta }\)-dense subset of \(X({\varOmega })\), but it is itself a \(G_{\delta }\)-dense set. We give an example of a domain in \(\mathbb {C}\) which is a \(H({\varOmega })\)-domain of holomorphy but not a \(A({\varOmega })\)-domain of holomorphy.