Runge Domain Research Articles

Flexible domains for minimal surfaces in Euclidean spaces

In this paper we introduce and investigate a new notion of flexibility for domains in Euclidean spaces Rn for n≥3 in terms of minimal surfaces which they contain. A domain Ω in Rn is said to be flexible if every conformal minimal immersion U→Ω from a Runge domain U in an open conformal surface M can be approximated uniformly on compacts, with interpolation on any given finite set, by conformal minimal immersion M→Ω. Together with hyperbolicity phenomena considered in recent works, this extends the dichotomy between flexibility and rigidity from complex analysis to minimal surface theory.

The *-product of domains in several complex variables

ABSTRACT In this article, we investigate the problem of computing the ∗ -product of domains in C N . Assuming that 0 ∈ G ⊂ C N is an arbitrary Runge domain and 0 ∈ D ⊂ C N is a bounded, smooth and linearly convex domain (or a non-decreasing union of such ones), we establish a geometric relation between D ∗ G and another domain in C N which is 'extremal' (in an appropriate sense) with respect to a special coefficient multiplier dependent only on the dimension N. Next, for N = 2, we derive a characterization of the latter domain expressed in terms of planar geometry. These two results, when combined together, give a formula which allows to calculate D ∗ G for two-dimensional domains D and G satisfying the outlined assumptions.

Shearing maps and a Runge map of the unit ball which does not embed into a Loewner chain with range Cn

In this paper we study the class of ``shearing'' holomorphic maps of the unit ball of the form $(z,w)\mapsto (z+g(w), w)$. Besides general properties, we use such maps to construct an example of a normalized univalent map of the ball onto a Runge domain in $\C^n$ which however cannot be embedded into a Loewner chain whose range is $\C^n$.

Polynomial hulls of arcs and curves

It is shown that there exist arcs and simple closed curves in C 3 \mathbb {C}^3 with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded, connected Runge domain of holomorphy in C N \mathbb {C}^N ( N ≥ 2 N \geq 2 ) there exist polynomially convex arcs and simple closed curves of almost full measure. These results, which strengthen earlier results of the author, are obtained as consequences of a general result about polynomial hulls of arcs and simple closed curves through compact, totally disconnected sets.

Complex curves in pseudoconvex Runge domains containing discrete subsets

For any pseudoconvex Runge domain Ω ⊂ ℂ2 we prove that every closed discrete subset in Ω is contained in a properly embedded complex curve in Ω with any prescribed topology (possibly infinite).

Runge tubes in Stein manifolds with the density property

In this paper we give a simple proof of the existence and plenitude of Runge tubes in C n \mathbb {C}^n ( n > 1 ) (n>1) and, more generally, in Stein manifolds with the density property. We show in particular that for any algebraic submanifold X X of codimension at least two in a complex Euclidean space C n \mathbb {C}^n , the normal bundle of X X admits a holomorphic embedding onto a Runge domain in C n \mathbb {C}^n which agrees with the inclusion map X ↪ C n X\hookrightarrow \mathbb {C}^n on the zero section.

A construction of complete complex hypersurfaces in the ball with control on the topology

Abstract Given a closed complex hypersurface Z ⊂ ℂ N + 1 {Z\subset\mathbb{C}^{N+1}} ( N ∈ ℕ {N\in\mathbb{N}} ) and a compact subset K ⊂ Z {K\subset Z} , we prove the existence of a pseudoconvex Runge domain D in Z such that K ⊂ D {K\subset D} and there is a complete proper holomorphic embedding from D into the unit ball of ℂ N + 1 {\mathbb{C}^{N+1}} . For N = 1 {N=1} , we derive the existence of complete properly embedded complex curves in the unit ball of ℂ 2 {\mathbb{C}^{2}} , with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc 𝔻 ⊂ ℂ {\mathbb{D}\subset\mathbb{C}} into the unit ball of ℂ 2 {\mathbb{C}^{2}} . These are the first known examples of complete bounded embedded complex hypersurfaces in ℂ N + 1 {\mathbb{C}^{N+1}} with any control on the topology.

Holomorphic functions on non-Runge domains and related problems

Let U be a domain in CN that is not a Runge domain. We study the topological and algebraic properties of the family of holomorphic functions on U which cannot be approximated by polynomials.

Rétraction capsulaire totale chez un patient avec un syndrome de Steinert

We prove that every complex analytic set X in a Runge domain Ω can be approximated by Nash sets on any relatively compact subdomain Ω0 of Ω. Moreover, for every Nash subset Y of Ω with Y⊂X, the approximating sets can be chosen so that they contain Y∩Ω0. As a consequence, we derive a necessary and sufficient condition for a complex analytic set X to admit a Nash approximation which coincides with X along its arbitrary given subset.

On Nash approximation of complex analytic sets in Runge domains

We prove that every complex analytic set X in a Runge domain Ω can be approximated by Nash sets on any relatively compact subdomain Ω0 of Ω. Moreover, for every Nash subset Y of Ω with Y⊂X, the approximating sets can be chosen so that they contain Y∩Ω0. As a consequence, we derive a necessary and sufficient condition for a complex analytic set X to admit a Nash approximation which coincides with X along its arbitrary given subset.

Approximation of holomorphic maps from Runge domains to affine algebraic varieties

We present a geometric proof of the theorem saying that holomorphic maps from Runge domains to affine algebraic varieties admit approximation by Nash maps. Next we generalize this theorem.

Approximation of Analytic Sets with Proper Projection by Algebraic Sets

Let X be an analytic subset of U×C n of pure dimension k such that the projection of X onto U is a proper mapping, where U⊂C k is a Runge domain. We show that X can be approximated by algebraic sets. Next we present a constructive method for local approximation of analytic sets by algebraic ones.

On closed invariant sets in local dynamics

We investigate the dynamical behaviour of a holomorphic map on an f-invariant subset C of U, where f : U → C k . We study two cases: when U is an open, connected and polynomially convex subset of C k and C ⋐ U , closed in U, and when ∂ U has a p.s.h. barrier at each of its points and C is not relatively compact in U. In the second part of the paper, we prove a Birkhoff's type theorem for holomorphic maps in several complex variables, i.e. given an injective holomorphic map f, defined in a neighborhood of U ¯ , with U star-shaped and f ( U ) a Runge domain, we prove the existence of a unique, forward invariant, maximal, compact and connected subset of U ¯ which touches ∂ U.

On a problem of Bremermann concerning Runge domains

In this paper we give an example of a bounded Stein domain in \(\mathbb{C}^{n}\), with smooth boundary, which is not Runge and whose intersection with every complex line is simply connected.

Approximation of analytic sets with proper projection by Nash sets

Let X be an analytic subset of U × C k of pure dimension such that the projection π : X → U is surjective and proper, where U is a Runge domain. We show that X can be approximated by Nash sets. To cite this article: M. Bilski, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Convolution Equations in spaces of Entire Functions of Exponential Type

In this paper, we present certain results on convolution equations and shift-invariant subspaces in special spaces of entire functions of exponential type. Problems related to a convolution equation and invariant subspaces were considered in various spaces of holomorphic functions (see, e.g., the surveys [1, 3] and the books [4, 5]), and the choice of the given spaces of holomorphic functions is related to the fact that in many other spaces of holomorphic functions, there arise difficulties connected with the harmonic analysis and synthesis, i.e., with the existence of invariant subspaces in which, as was proved by D. I. Gurevich [10], either there are no polynomial–exponential element, or they are not dense in a subspace. In the chosen class, there are none of these initial difficulties. 1. Space of Entire Functions PG Let G be a Runge domain in the space C n , O(G) be the space of functions holomorphic in G equipped with the natural topology, i.e., the topology of the uniform convergence on all compact sets in G. Denote by O � (G) the dual space to O(G), i.e., the space of all linear continuous functionals on O(G )e quipped with the strong topology. Sometimes, elements of T ∈O � (G) are called analytic functionals on the domain G. The space O(G) is a closed subspace of the space C 0 (G) of all functions continuous in G, and, therefore, for every analytic functional T ∈O � (G), there exists a measure µ, supp µ G, such that

Complex Mean-Value Interpolation and Approximation of Holomorphic Functions

We show that complex mean-value interpolation, a generalizationof Lagrange–Hermite interpolation, may be defined in anydomain that is C -convex, whereas the original definition requiredordinary, real convexity. We also show that C -convex domainsare the natural ones in which to perform mean-value interpolation,in the sense that any Runge domain which admits mean-value interpolationmust in fact be C -convex. Finally, we obtain an integral formulafor the error and give some applications concerning approximationof holomorphic functions.

Polynomial Convexity of Teichmüller Spaces

It is well known that finite-dimensional Teichmüller spaces are holomorphically convex, that is, they are domains of holomorphy. Moreover, the holomorphic convexity is fulfilled for all Teichmüller spaces in a stronger form: they are complex hyperconvex. In this note we establish that, in fact, finite-dimensional Teichmüller spaces possess a much stronger convexity property, namely, they are polynomially convex; in other words, they are Runge domains. Additionally, some geometric properties of Teichmüller spaces are established.

Algebraic approximations of holomorphic maps from Stein domains to projective manifolds

It is shown that every holomorphic map $f$ from a Runge domain $\Omega$ of an affine algebraic variety $S$ into a projective algebraic manifold $X$ is a uniform limit of Nash algebraic maps $f_\nu$ defined over an exhausting sequence of relatively compact open sets $\Omega_\nu$ in $\Omega$. A relative version is also given: If there is an algebraic subvariety $A$ (not necessarily reduced) in $S$ such that the restriction of $f$ to $A\cap\Omega$ is algebraic, then $f_\nu$ can be taken to coincide with $f$ on $A\cap\Omega_\nu$. The main application of these results, when $\Omega$ is the unit disk, is to show that the Kobayashi pseudodistance and the Kobayashi-Royden infinitesimal metric of a quasi-projective algebraic manifold $Z$ are computable solely in terms of the closed algebraic curves in $Z$. Similarly, the $p$-dimensional Eisenman metric of a quasi-projective algebraic manifold can be computed in terms of the Eisenman volumes of its $p$-dimensional algebraic subvarieties. Another question addressed in the paper is whether the approximations $f_\nu$ can be taken to have their images contained in affine Zariski open subsets of $X$. By using complex analytic methods (pluricomplex potential theory and Hormander's $L^2$ estimates), we show that this is the case if $f$ is an embedding (with $\dim S<\dim X$) and if there is an ample line bundle $L$ on $X$ such that

Complements of Runge domains and holomorphic hulls.

Complements of Runge domains and holomorphic hulls.