Variations of complex structures on an open Riemann surface

M. S. Narasimhan

Displaying similar documents to “Variations of complex structures on an open Riemann surface”

Holomorphic series expansion of functions of Carleman type

Taib Belghiti (2004)

Annales Polonici Mathematici

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Let f be a holomorphic function of Carleman type in a bounded convex domain D of the plane. We show that f can be expanded in a series f = ∑ₙfₙ, where fₙ is a holomorphic function in Dₙ satisfying $s u p_{z \in D ₙ} | f ₙ (z) | \leq C ϱ ⁿ$ for some constants C > 0 and 0 < ϱ < 1, and where (Dₙ)ₙ is a suitably chosen sequence of decreasing neighborhoods of the closure of D. Conversely, if f admits such an expansion then f is of Carleman type. The decrease of the sequence Dₙ characterizes the smoothness of f. ...

A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces

Viêt-Anh Nguyên (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$ ) be a non locally pluripolar subset of $X$ (resp. $Y$ ). We show that every separately holomorphic mapping $f : W : = (A \times Y) \cup (X \times B) \to Z$ extends to a holomorphic mapping $\hat{f}$ on $\hat{W} : = \{(z, w) \in X \times Y : \tilde{ω} (z, A, X) + \tilde{ω} (w, B, Y) < 1\}$ such that $\hat{f} = f$ on $W \cap \hat{W},$ where $\tilde{ω} (\cdot, A, X)$ (resp. $\tilde{ω} (\cdot, B, Y))$ is the plurisubharmonic measure of $A$ (resp. $B$ ) relative to $X$ (resp. $Y$ ). Generalizations...

A boundary cross theorem for separately holomorphic functions

Peter Pflug, Viêt-Anh Nguyên (2004)

Annales Polonici Mathematici

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Let D ⊂ ℂⁿ and $G \subset ℂ^{m}$ be pseudoconvex domains, let A (resp. B) be an open subset of the boundary ∂D (resp. ∂G) and let X be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Suppose in addition that the domain D (resp. G) is locally ² smooth on A (resp. B). We shall determine the “envelope of holomorphy” X̂ of X in the sense that any function continuous on X and separately holomorphic on (A×G)∪(D×B) extends to a function continuous on X̂ and holomorphic on the interior of X̂. A generalization of this...

Analytic cohomology of complete intersections in a Banach space

Imre Patyi (2004)

Annales de l’institut Fourier

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Let $X$ be a Banach space with a countable unconditional basis (e.g., $X = ℓ_{2}$ ), $Ω \subset X$ an open set and $f_{1}, ..., f_{k}$ complex-valued holomorphic functions on $Ω$ , such that the Fréchet differentials $d f_{1} (x), ..., d f_{k} (x)$ are linearly independant over $ℂ$ at each $x \in Ω$ . We suppose that $M = {x \in Ω : f_{1} (x) = ... = f_{k} (x) = 0}$ is a complete intersection and we consider a holomorphic Banach vector bundle $E \to M$ . If $I$ (resp. $𝒪^{E}$ ) denote the ideal of germs of holomorphic functions on $Ω$ that vanish on $M$ (resp. the sheaf of germs of holomorphic sections of $E$ ), then the sheaf cohomology groups...

Regular and limit sets for holomorphic correspondences

S. Bullett, C. Penrose (2001)

Fundamenta Mathematicae

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Holomorphic correspondences are multivalued maps $f = Q ̃ ₊ Q ̃ ₋^{- 1} : Z \to W$ between Riemann surfaces Z and W, where Q̃₋ and Q̃₊ are (single-valued) holomorphic maps from another Riemann surface X onto Z and W respectively. When Z = W one can iterate f forwards, backwards or globally (allowing arbitrarily many changes of direction from forwards to backwards and vice versa). Iterated holomorphic correspondences on the Riemann sphere display many of the features of the dynamics of Kleinian groups and rational maps,...

Zeros of bounded holomorphic functions in strictly pseudoconvex domains in $ℂ^{2}$

Jim Arlebrink (1993)

Annales de l'institut Fourier

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Let $D$ be a bounded strictly pseudoconvex domain in $ℂ^{2}$ and let $X$ be a positive divisor of $D$ with finite area. We prove that there exists a bounded holomorphic function $f$ such that $X$ is the zero set of $f$ . This result has previously been obtained by Berndtsson in the case where $D$ is the unit ball in $ℂ^{2}$ .

On the Rogosinski radius for holomorphic mappings and some of its applications

Lev Aizenberg, Mark Elin, David Shoikhet (2005)

Studia Mathematica

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The well known theorem of Rogosinski asserts that if the modulus of the sum of a power series is less than 1 in the open unit disk: $| \sum_{n = 0}^{\infty} a ₙ z ⁿ | < 1$ , |z| < 1, then all its partial sums are less than 1 in the disk of radius 1/2: $| \sum_{n = 0}^{k} a ₙ z ⁿ | < 1$ , |z| < 1/2, and this radius is sharp. We present a generalization of this theorem to holomorphic mappings of the open unit ball into an arbitrary convex domain. Other multidimensional analogs of Rogosinski’s theorem as well as some applications to dynamical systems are...

A set on which the local Łojasiewicz exponent is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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Let U be a neighbourhood of 0 ∈ ℂⁿ. We show that for a holomorphic mapping $F = (f ₁, . . ., f ₘ) : U \to ℂ^{m}$ , F(0) = 0, the Łojasiewicz exponent ₀(F) is attained on the set z ∈ U: f₁(z)·...·fₘ(z) = 0.

A result on extension of C.R. functions

Makhlouf Derridj, John Erik Fornaess (1983)

Annales de l'institut Fourier

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Let $Ω$ an open set in $C^{4}$ near $z_{0} \in \partial Ω$ , $λ$ a suitable holomorphic function near $z_{0}$ . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\overline{\partial} u = λ f$ , ( $f$ is a $(0, 1)$ form, $\overline{\partial}$ closed in $U (z_{0})$ in $U (z_{0})$ with supp $(u) \subset \overline{Ω} \cap U (z_{0})$ , then we deduce an extension result for $C . R .$ functions on $\partial Ω \cap U (z_{0})$ , as holomorphic fonctions in $Ω \cap V (z_{0})$ .

Contracting rigid germs in higher dimensions

Matteo Ruggiero (2013)

Annales de l’institut Fourier

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Following Favre, we define a holomorphic germ $f : (ℂ^{d}, 0) \to (ℂ^{d}, 0)$ to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of $f$ and its linear action on the fundamental group of the complement of the critical set.