A schlichtness theorem for envelopes of holomorphy

A note on envelopes of holomorphy

Let $p:X\longrightarrow M$ be a Riemann domain over a connected $n$-dimensional complex submanifold $M$ of $\mathbb C^N$ and let $\mathcal F\subset\mathcal O(X)$ be such that $p\in\mathcal F^N$. Our aim is to discuss relations between the $\mathcal F$-envelope of holomorphy of $(X,p)$ in the sense of Riemann domains over $M$ and the $\mathcal F$-envelope of holomorphy of $X$ in the sense of complex manifolds.

Envelope of holomorphy for boundary cross sets

Let $D\subset \C^n,$ $G\subset \C^m$ be open sets, let $A$ (resp. $B$) be a subset of the boundary $\partial D$ (resp. $\partial G$) and let $W$ be the 2-fold boundary cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ An open subset $X\subset\C^{n+m} $ is said to be the ``envelope of holomorphy" of $W$ if it is, in some sense, the maximal open set with the following property: Any function locally bounded on $W$ and separately holomorphic on $(A\times G) \cup (D\times B)$ "extends" to a holomorphic function defined on $X$ which admits the boundary values $f$ a.e. on $W.$ In this work we will …

Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space

In this paper we investigate invariant domains in $\, Ξ^+$, a distinguished $\,G$-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space $\,G/K$. The domain $\,Ξ^+$, recently introduced by Krötz and Opdam, contains the crown domain $\,Ξ\,$ and it is maximal with respect to properness of the $\,G$-action. In the tube case, it also contains $\,S^+$, an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of $\,Ξ$. We prove that the envelope of holomorphy of an invariant domain in $\,Ξ^+$, which is contained n…

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a nei…

Envelopes of holomorphy and extension of functions of bounded type

We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not…

高维复空间中截断管状域与特殊域的包络

Envelope of truncated tubes and special domains in higher complex dimensions

本文引入特殊域并证明拟凸截断管状域均为特殊域，进而将 Jarnicki-Pflug 关于截断管状域全纯包络的结果推广到 $\C^n$ 中特殊域的包络，同时给出复二维单叶性结果及其两个高维推广。

A new Hartogs type extension results for the cross-like objects

We discuss the problem of the existence of envelopes of holomorphy of the A-crosses, which leads us to the far-reaching generalizations of the famous Hartogs theorem. We also take under consideration the issue of the existence of "nice" general descriptions of envelopes of holomorphy of the cross-like objects in terms of the relative extremal function, which seems to be very natural in the light of the extension theory of separately holomorphic functions on classical crosses and (N,k)-crosses.

Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions; Tumanov's solution of Bishop's equation in Hoelder classes with optimal loss of smoothness; wedge-extendability on C^2,a generic submanifolds of C^n consisting of a single CR orbit; propagation of CR extendability and edge-of-the-wedge theorem; Painlevé problem; metrically thin …

复 Grassmann 流形中 Matsuki 轨道的全纯包络

Envelope of Holomorphy of Matsuki Orbits in Complex Grassmannians

本文利用 Rossi 全纯延拓理论与全纯纤维丛结构，证明复 Grassmann 流形中 Matsuki 轨道的全纯包络双全纯等价于包含它的 $K$-轨道。

On envelopes of holomorphy of domains covered by Levi-flat hats and the reflection principle

In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat ``hat''. In our main theorem, we show that every C^infty-smooth CR diffeomorphism h: M to M' between two globally minimal real analytic hypersurfaces in C^n (n >1) is real analytic at every point of M if M' is holomorphically nondegenerate. More generally, we establish th…

Characteristic foliations on maximally real submanifolds of C^n and envelopes of holomorphy

Let S be an arbitrary real surface, with or without boundary, contained in a hypersurface M of the complex euclidean space \C^2, with S and M of class C^{2, a}, where 0 < a < 1. If M is globally minimal, if S is totally real except at finitely many complex tangencies which are hyperbolic in the sense of E. Bishop and if the union of separatrices is a tree of curves without cycles, we show that every compact K of S is CR-, W- and L^p-removable (Theorem~1.3). We treat this seemingly global problem by means of purely local techniques, namely by means of families of small analytic discs partially …

Cross theorem

Let $D, G\subset{\Bbb C}$ be domains, let $A\subset D$, $B\subset G$ be locally regular sets, and let $X:=(D\times B)\cup(A\times G)$. Assume that $A$ is a Borel set. Let $M$ be a proper analytic subset of an open neighborhood of $X$. Then there exists a pure 1-dimensional analytic subset $\hat M$ of the envelope of holomorphy $\hat X$ of $X$ such that any function separately holomorphic on $X\setminus M$ extends to a holomorphic function on $\hat X\setminus\hat M$. The result generalizes special cases which were studied in \cite{Ökt 1998}, \cite{Ökt 1999a}, and \cite{Sic 2000}.

A boundary cross theorem for separately holomorphic functions

Let $D\subset \C^n,$ $G\subset \C^m$ be pseudoconvex domains, let $A$ (resp. $B$) be an open subset of the boundary $\partial D$ (resp. $\partial G$) and let $X$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Suppose in addition that the domain $D$ (resp. $G$) is {\it locally $\mathcal{C}^2$ smooth on $A$} (resp. $B$). We shall determine the "envelope of holomorphy" $\hat{X}$ of $X$ in the sense that any function continuous on $X$ and separately holomorphic on $(A\times G) \cup (D\times B)$ extends to a function continuous on $\hat{X}$ and holomorphic on the interior of $\ha…

A brief proof of Bochner's tube theorem and a generalized tube

The aim of this note is firstly to give a new brief proof of classical Bochner's Tube Theorem (1938) by making use of K. Oka's Boundary Distance Theorem (1942), showing directly that two points of the envelope of holomorphy of a tube can be connected by a line segment. We then apply the same idea to show that if an unramified domain $\mathfrak{D}:=A_1+iA_2 \to \mathbf{C}^n$ with unramified real domains $A_j \to \mathbf{R}^n$ is pseudoconvex, then the both $A_j$ are univalent and convex (a generalization of Kajiwara's theorem). From the viewpoint of this result we discuss a generalization by M.…

Boundary cross theorem in dimension 1

Let $X, Y$ be two complex manifolds of dimension 1 which are countable at infinity, let $D\subset X,$ $ G\subset Y$ be two open sets, let $A$ (resp. $B$) be a subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Suppose in addition that $D$ (resp. $G$) is {\it Jordan-curve-like on $A$} (resp. $B$) and that $A$ and $B$ are {\it of positive length}. We determine the "envelope of holomorphy" $\hat{W}$ of $W$ in the sense that any function locally bounded on $W,$ measurable on $A\times B,$ and separately holomorphic on $(A\times…

Boundary cross theorem in dimension 1 with singularities

Let $D$ and $G$ be copies of the open unit disc in $\C,$ let $A$ (resp. $B$) be a measurable subset of $\partial D$ (resp. $\partial G$), let $W$ be the 2-fold cross $\big((D\cup A)\times B\big)\cup \big(A\times(B\cup G)\big),$ and let $M$ be a relatively closed subset of $W.$ Suppose in addition that $A$ and $B$ are of positive one-dimensional Lebesgue measure and that $M$ is fiberwise polar (resp. fiberwise discrete) and that $M\cap (A\times B)=\varnothing.$ We determine the "envelope of holomorphy" $\hat{W\setminus M}$ of $W\setminus M$ in the sense that any function locally bounded on $W\s…

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let $\,G/K\,$ be a non-compact irreducible Hermitian symmetric space of rank $\,r\,$ and let $\,NAK\,$ be an Iwasawa decomposition of $\,G$. By the polydisc theorem, $\,AK/K\,$ can be regarded as the base of an $\,r$-dimensional tube domain holomorphically embedded in $\,G/K$. As every $\,N$-orbit in $\,G/K\,$ intersects $\,AK/K$ in a single point, there is a one-to-one correspondence between $\,N$-invariant domains in $\,G/K\,$ and tube domains in the product of $\,r\,$ copies of the upper half-plane in $\,\C$. In this setting we prove a generalization of Bochner's tube theorem. Namely, an $\…

On the Hartogs extension theorem for unbounded domains in $\mathbb{C}^n$

Let $Ω\subset\mathbb{C}^n$, $n\geq 2$, be a domain with smooth connected boundary. If $Ω$ is relatively compact, the Hartogs-Bochner theorem ensures that every CR distribution on $\partialΩ$ has a holomorphic extension to $Ω$. For unbounded domains this extension property may fail, for example if $Ω$ contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of $\mathbb{C}^n\backslash\overlineΩ$ is $\mathbb{C}^n$. It seems that it is a first result in the literature which gives a geometric characterization of…

An extension theorem for separately holomorphic functions with singularities

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb C^{k_1+...+k_N}.$$Let $U$ be an open connected neighborhood of $X$ and let $M\varsubsetneq U$ be an analytic subset. Then there exists an analytic subset $\hat M$ of the `envelope of holomorphy' $\hat X$ of $X$ with $\hat M\cap X\subset M$ such that for every function $f$ separately holomorphic on $X\setminus M$ there exists an $\hat f$ holomorphic on $\hat X\setminus\hat…

An extension theorem for separately meromorphic functions with pluripolar singularities

Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset X$ be relatively closed. For any $j\in\{1,...,N\}$ let $Σ_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1})\times(A_{j+1}\times...\times A_N)$ such that the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb C^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $Σ_1,...,Σ_N$ are pluripolar. Put ${multline*} X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1…

An extension theorem for separately holomorphic functions with pluripolar singularities

Let $D_j\subset\Bbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset\Bbb C^{n_1}\times...\times\Bbb C^{n_N}=\Bbb C^n. $$ Let $U\subset\Bbb C^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,...,N\}$ let $Σ_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1}) \times(A_{j+1}\times...\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\Bbb C^{n_j}\: (z',z_j,z'')\in M…