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 determine the envelope of holomorphy of some boundary crosses.
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.
Let Omega be a domain in C^2. We prove the following theorem. If the envelope of holomorphy of Omega is schlicht over Omega, then the envelope is in fact schlicht. We provide examples showing that the theorem is not true in C^n, n>2. Additionally, we show that the theorem cannot be generalized to provide information about domains in C^2 whose envelopes are multiply sheeted.
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…
Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain Y in X an interval domain with respect to F if Y intersects every leaf of F in a nonempty connected set. We show that if X is Stein and if F satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope…
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…
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…
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.
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 …
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…
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 …
We give two sufficient criteria for schlichtness of envelopes of holomorphy in terms of topology. These are weakened converses of results of Kerner and Royden. Our first criterion generalizes a result of Hammond in dimension 2. Along the way we also prove a generalization of Royden's 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}.
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…
Let $X, Y$ be two complex manifolds, let $D\subset X,$ $ G\subset Y$ be two nonempty open sets, let $A$ (resp. $B$) be an open subset of $\partial D$ (resp. $\partial G$), and let $W$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Under a geometric condition on the boundary sets $A$ and $B,$ we show that every function locally bounded, separately continuous on $W,$ continuous on $A\times B,$ and separately holomorphic on $(A\times G) \cup (D\times B)$ "extends" to a function continuous on a "domain of holomorphy" $\hat{W}$ and holomorphic on the interior of $\hat{W}.$
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…
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…
We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of separable spaces.
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…
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…
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…
Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${\mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a domain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f \, (\, =(f_1,f_2,\cdots,f_n)\, )$ is a $\mathbb C^n$-valued function on an open set $U$ of $\mathbb C^{k_1}\times\mathbb C^{k_2}\times\cdots\times\mathbb C^{k_n}$, $f$ is said to be $\mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i\not=…
On boundary points of minimal continuously Hutchinson invariant sets
Stockholm University（斯德哥尔摩大学）；Stony Brook University（石溪大学）；Weizmann Institute of Science（魏茨曼科学研究所）；Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）
