Erlend Fornæss Wold
We show that any finitely connected domain $U\subset\CC$ can be properly embedded into $\CC^2$. For some sequences $\{p_j\}\subset U$, $U\setminus\{p_j\}$ can also be properly embedded into $\CC^2$.
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A space $G(M, \varPhi)$ of infinitely differentiable functions in ${\mathbb R}^n$ constructed with a help of a family $\varPhi=\{φ_m\}_{m=1}^{\infty}$ of real-valued functions $φ_m \in~C({\mathbb R}^n)$ and a logarithmically convex sequence $M$ of positive numbers is considered in the article. In view of conditions on $M$ each function of $G(M, \varPhi)$ can be extended to an entire function in ${\mathbb C}^n$. Imposed conditions on $M$ and $\varPhi$ allow to describe the space of such extensions.
Holomorphic mappings of finitely connected planar domains
本文探究无需Koebe定理证明有限连通平面区域全纯映射相关结果，给出连通度>2的有限连通区域间真全纯映射的刚性定理，还提供不依赖欧拉示性数的平面Riemann-Hurwitz公式新初等证明。
Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map $X \to \C\times\C^*$ from a finitely connected planar domain $X$ without isolated boundary points, a stronger Oka property holds, namely that the map is homotopic to a proper holomorphic embedding. Here we show that every continuous map from a finitely connected planar domain, possibly with punctures, into $\C\times\C^*$ is homotopic to a proper immersion that identifies …
We develop a theory of holomorphic differentials on a certain class of non-compact Riemann surfaces obtained by opening infinitely many nodes.
The algebra $H^\infty(D)$ of bounded holomorphic functions on $D\subset\mathbb C$ is projective free for a wide class of infinitely connected domains. In particular, for such $D$ every rectangular left-invertible matrix with entries in $H^\infty(D)$ can be extended in this class of matrices to an invertible square matrix (the generalization of the corona theorem for $H^\infty(D)$). This follows from a new result on the structure of the maximal ideal space of $H^\infty(D)$ asserting that its covering dimension is $2$ and the second Čech cohomology group is trivial.
We consider a space of infinitely smooth functions on an unbounded closed convex set in ${\mathbb R}^n$. It is shown that each function of this space can be extended to an entire function in ${\mathbb C}^n$ satisfying some prescribed growth condition. Description of linear continuous functionals on this space in terms of their Fourier-Laplace transform is obtained. Also a variant of the Paley-Wiener-Schwartz theorem for tempered distributions is given it the paper.
In this paper, we present a new technique for studying the dynamics of a finitely generated rational semigroup. Such a semigroup can be associated naturally to a certain holomorphic correspondence on $\mathbb{P}^1$. Then, results on the iterative dynamics of such a correspondence can be applied to the study of the rational semigroup. We focus on a certain invariant measure for the aforementioned correspondence---known as the equilibrium measure. This confers some advantages over many of the known techniques for studying the dynamics of rational semigroups. We use the equilibrium measure to ana…
We prove that every circled domain in the Riemann sphere admits a proper holomorphic embedding to C^2. Our methods also apply to circled domains with punctures, provided that all but finitely many of the punctures belong to the closure of the set of complementary discs.
Let D be a bounded, finitely connected domain in the complex plane without isolated points in the boundary and let f be a continuous function on the boundary bD. Let F be a continuous extension of f to the closure of D. We prove that f extends holomorphically through D if and only if the degree of F+h is nonnegative for every holomorphic function h on D such that F+h is bounded away from zero near bD.
We give a complete description of bounded Reinhardt domains of finite boundary smoothness that have non-compact automorphism group. As part of this program, we show that the classification of domains with non-compact automorphism group and having only finite boundary smoothness is considerably more complicated than the classification of such domains that have infinitely smooth boundary.
It is shown that if the boundary of a Reinhardt domain in $\mathbb{C}^n$ contains the origin, each holomorphic function on the domain which is infinitely many times differentiable up to the boundary extends holomorphically to a neighborhood of the origin.
Let D be a bounded domain in the complex plane whose boundary consists of finitely many pairwise disjoint simple closed curves. Give bD the standard orientation and let A(D) be the algebra of all continuous functions on the closure of D which are holomorphic on D. In the paper we prove that a continuous function f on bD extends to a function in A(D) if and only if for each g in A(D) such that f+g has no zero on bD the change of argument of f+g along bD is nonnegative.
Holomorphic maps sharing preimages over finitely generated fields
本文研究紧黎曼曲面上全纯映射在有限生成域上共享原像的条件，证明存在无限完全不变集时两映射为亏格0或1曲面全纯伽罗瓦覆盖的合成左因子，并考虑更一般的原像相等方程。
Rigidity of symmetric holomorphic functions on infinite-dimensional sequence spaces
本文研究无限维序列空间上置换不变全纯函数，证明在链连通域上对称性导致刚性（常值性），并给出边界反例与向量值推论。
The main purpose of the present paper is to introduce the notion of squeezing functions of bounded domains and study some properties of them. The relation to geometric and analytic structures of bounded domains will be investigated. Existence of related extremal maps and continuity of squeezing functions are proved. Holomorphic homogeneous regular domains are exactly domains whose squeezing functions have positive lower bounds. Completeness of certain intrinsic metrics and pseudoconvexity of holomorphic homogeneous regular domains are proved by alternative method. In dimension one case, we get…
We show that every holomorphic map $f\in\mathcal{H}(Ω\setminus K)$ ($K\subsetΩ\subset\mathbb{C}^n$, with $K$ compact, $Ω$ open, and $n\ge2$), has a unique "\emph{Hartogs companion}" $\tilde f\in\mathcal{H}(Ω)$ matching $f$ on an open subset $C_{K,Ω}\subsetΩ\setminus K$. Furthermore, $\tilde f$ extends $f$, \emph{if and only if} $\mathbb{C}^n\setminus K$ is a connected set; this equivalence proves the converse implication from the Hartogs Kugelsatz. The existence of vector-valued Hartogs companions in any dimension yields a Hartogs-type extension theorem for Gâteaux holomorphic maps $f\in\mathc…
In this paper, we explore holomorphic Segre preserving maps. First, we investigate holomorphic Segre preserving maps sending the complexification $\mathcal{M}$ of a generic real analytic submanifold $M \subseteq \C^N$ of finite type at some point $p$ into the complexification $\mathcal{M}'$ of a generic real analytic submanifold $M' \subseteq \C^{N'}$, finitely nondegenerate at some point $p'$. We prove that for a fixed $M$ and $M'$, the germs at $(p,\bar{p})$ of Segre submersive holomorphic Segre preserving maps sending $(\M,(p,\bar{p}))$ into $(\M',(p', \bar{p}'))$ can be parametrized by the…
We prove that any Lie subgroup $G$ (with finitely many connected components) of an infinite-dimensional topological group $\mathcal G$ which is an amalgamated product of two closed subgroups, can be conjugated to one factor. We apply this result to classify Lie group actions on Danielewski surfaces by elements of the overshear group (up to conjugation).
We extend and simplify results of \cite{Din~2009} where the asymptotic behavior of the holomorphic sectional curvature of the Bergman metric in annuli is studied. Similarly as in \cite{Din~2009} the description enables us to construct an infinitely connected planar domain (in our paper it is a Zalcman type domain) for which the supremum of the holomorphic sectional curvature is two whereas its infimum is equal to $-\infty$.
We consider the number of zeros of holomorphic functions in a bounded domain that depend on a small parameter and satisfy an exponential upper bound near the boundary of the domain and similar lower bounds at finitely many points along the boundary. Roughly the number of such zeros is $(2πh)^{-1}$ times the integral over the domain of the laplacian of the exponent of the dominating exponential. Such results have already been obtained by M. Hager and by Hager and the author and they are of importance when studying the asymptotic distribution of eigenvalues of elliptic operators with small rando…
We prove that the Bergman kernel function associated to a finitely connected domain in the plane is given as a rational combination of only three basic functions of one complex variable: an Alhfors map, its derivative, and one other function whose existence is deduced by means of the field of meromorphic functions on the double of the domain. Because many other functions of conformal mapping and potential theory can be expressed in terms of the Bergman kernel, our results shed light on the complexity of these objects. We also prove that the Bergman kernel is an algebraic function of a single A…
Dynamics of composition operators on spaces of holomorphic functions on plane domains
本文刻画了平面域上全纯函数空间中加权复合算子的超循环性与拓扑混合性之间的等价关系，并针对穿孔平面、穿孔圆盘及有限连通域等情形给出了超循环性的完整判据。
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