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
$f: W:=(A\times Y) \cup (X\times B)\longrightarrow Z$ extends to a holomorphic mapping $\hat{f}$ on $\hat{W}:=\left\lbrace(z,w)\in X\times Y:\ \widetildeω(z,A,X)+\widetildeω(w,B,Y)<1 \right\rbrace$ such that $\hat{f}=f$ on $W\cap \hat{W},$ where $\widetildeω(\cdot,A,X)$ (resp. $\widetildeω(\cdot,B,Y))$ is the plurisubharmonic measure of $A$ (resp. $B$) relative to $X$ (resp. $Y$).
Generalizations of this result for an $N$-fold cross are also given.
Answering all questions---concerning proper holomorphic mappings between generalized Hartogs triangles---posed by Jarnicki and Plfug (First steps in several complex variables: Reinhardt domains, 2008) we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, we completely describe the group of holomorphic automorphisms of such domains and establish rigidity of proper holomorphic self-mappings on them.
The explicit form of proper holomorphic mappings between complex ellipsoids is given. Using this description, we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, the automorphism group of such domains is found.
A generalized Fock-Bargmann-Hartogs domain $D_n^{\mathbf{m},\mathbf{p}}$ is defined as a domain fibered over $\mathbb{C}^{n}$ with the fiber over $z\in \mathbb{C}^{n}$ being a generalized complex ellipsoid $Σ_z({\mathbf{m},\mathbf{p}})$. In general, a generalized Fock-Bargmann-Hartogs domain is an unbounded non-hyperbolic domains without smooth boundary. The main contribution of this paper is as follows. By using the explicit formula of Bergman kernels of the generalized Fock-Bargmann-Hartogs domains, we obtain the rigidity results of proper holomorphic mappings between two equidimensional gen…
The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the $L^p$ mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.
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…
Consider a domain $\varOmega$ in $\mathbb{C}^n$ with $n\geqslant 2$ and a compact subset $K\subset\varOmega$ such that $\varOmega\backslash K$ is connected. We address the problem whether a holomorphic line bundle defined on $\varOmega\backslash K$ extends to $\varOmega$. In 2013, Fornæss, Sibony and Wold gave a positive answer in dimension $n\geqslant 3$, when $\varOmega$ is pseudoconvex and $K$ is a sublevel set of a strongly plurisubharmonic exhaustion function. However, for $K$ of general shape, we construct counterexamples in any dimension $n\geqslant 2$. The key is a certain gluing lemma…
In this paper, we characterize the Hartogs domains over homogeneous Siegel domains of type II and explicitly describe their automorphism groups. Moreover we prove that any proper holomorphic map between Hartogs domains over homogeneous Siegel domains over type II is a biholomorphism.
The aim of this paper is to present an extension theorem for the functions separately holomorphic on generalized (N,k)-crosses with pluripolar singularities.
Sobolev irregularity of the Bergman projection on a family of domains containing the Hartogs triangle is shown. On the Hartogs triangle itself, a sub-Bergman projection is shown to satisfy better Sobolev norm estimates than its Bergman projection.
In this paper we present a proof of Hartogs' extension theorem, following T. Sobieszek's paper from 2003. Hartogs' theorem provides a large class of domains where holomorphic functions have analytic continuation to larger domains, and is "a several complex variables theorem" in nature because its conclusion is false in the complex plane. Sobieszek's proof is quite remarkable because he uses, stated in his paper without proofs, only higher-dimensional identity principle for holomorphic functions and Cauchy's integral formula for compact sets. We proved this two theorems here, making this exposi…
We prove an extension theorem for roots and logarithms of holomorphic line bundles across strictly pseudoconcave boundaries: they extend in all cases except one, when dimension and Morse index of a critical point is two. In that case we give an explicit description of obstructions to the extension.
We generalize Chirka's theorem on the extension of functions holomorphic in a neighbourhood of graph(F)\cup(\partial D\times D) -- where D is the open unit disc and graph(F) denotes the graph of a continuous D-valued function F -- to the bidisc. We extend holomorphic functions by applying the Kontinuitaetssatz to certain continuous families of analytic annuli, which is a procedure suited to configurations not covered by Chirka's theorem.
Let D be a Hartogs domain of the form D={(z,w) \in CxC^N : |w| < e^{-u(z)}} where u is a subharmonic function on C. We prove that the Bergman space of holomorphic and square integrable functions on D is either trivial or infinite dimensional.
Let $Ω$ be a bounded pseudoconvex Hartogs domain. There exists a natural complete Kähler metric $g^Ω$ in terms of its defining function. In this paper, we study two problems. The first one is determining when $g^Ω$ is Einstein or extremal. The second one is the existence of holomorphic isometric immersions of $(Ω, g^Ω)$ into finite or infinite dimensional complex space forms.
We prove an analog of the classical Hartogs extension theorem for CR $L^{2}$ functions defined on boundaries of certain (possibly unbounded) domains on coverings of strongly pseudoconvex manifolds. Our result is related to a problem posed in the paper of Gromov, Henkin and Shubin [GHS] on holomorphic $L_{2}$ functions on coverings of pseudoconvex manifolds.
We prove the following Hartogs-Bochner type theorem: Let $M$ be a connected $C^2$ hypersurface of $P_n(\mathbb{C})$ ($n\geq 2$) which divides $P_n(\mathbb{C})$ in two connected open sets $Ω_1$ and $Ω_2$. Then there exists $i \in \{1,2\}$ such that $C^{1}$ CR functions defined on $M$ extends holomorphically to $Ω_i$.
In this paper we prove a metric version of Hartogs' theorem where the holomorphic function is replaced by a locally symmetric Hermitian metric. As an application, we prove that if the Kobayashi metric on a strongly pseudoconvex domain with $\mathcal{C}^2$ smooth boundary is a Kähler metric, then the universal cover of the domain is the unit ball.
For a large class of convex domains in $\bf C^n$, it is shown that an $L^p$ function on the boundary is CR if there are holomorphic extensions on almost all slices of D by complex lines parallel to the coordinate axes. As an application, a new operator limit for the Szegő projection is obtained.
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.
The present paper have been replaced by the paper: Hartogs-Bochner type theorem in Projective Space (math.CV/0011095) in which we prove the following Hartogs-Bochner type theorem: Let $M$ be a connected $C^2$ hypersurface of $P_n(\mathbb{C})$ ($n\geq 2$) which divides $P_n(\mathbb{C})$ in two connected open sets $Ω_1$ and $Ω_2$. Then there exists $i \in \{1,2\}$ such that $C^{1}$ CR functions defined on $M$ extends holomorphically to $Ω_i$.
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbf{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbf{C}^n\times \mathbf{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbf{C}^{n+m}$. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock-Bargmann-Hartogs domains in terms of the polylogarithm functions and Kim-Ninh-Yamamori determined the automorphism group of the domain $D_{n,m}(μ)$. In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock-Bargmann-Hartogs …
Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D such that D - K is connected, holomorphic or meromorphic functions in D - K extend holomorphically or meromorphically to D. Normality is an unvavoidable assumption for holomorphic extension, but we show that meromorphic extension holds on a reduced globally irreducible (not neces…
100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was stated in full generality later by Osgood (1929): holomorphic functions in a connected neighborhood V(bD) of a connected boundary bD contained in C^n (n >= 2) do extend holomorphically and uniquely to the domain D. It was a long-standing open problem to derive a proof using only analytic discs, as did Hurwitz (1897), Hartogs (1906) and E.E. Levi (1911) in some special, model cases. Quite unexpectedly, Fornaess in 1998 exhibited a topologically s…
Let M be a compact, connected, C^2-smooth and globally minimal hypersurface M in P_2(C) which divides the projective space into two connected parts U^{+} and U^{-}. We prove that there exists a side, U^- or U^+, such that every continuous CR function on M extends holomorphically to this side. Our proof of this theorem is a simplification of a result originally due to F. Sarkis.
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…
A question of Poletsky was to know if there exists a thin Hartogs figure such that any of its neighborhoods cannot be imbedded in Stein spaces. In \cite{chirka}, Chirka and Ivashkovitch gave such an example arising in an open complex manifold. In this paper, we answer to the question of the existence of such a figure in compact surfaces by giving an example arising in $P_2(\mathbb{C})$. By smoothing it, we obtain a smooth (non analytic) disc with boundary $\bar{D} \subset P_2(\mathbb{C})$ having the same property. Consequently, this disc intersects all algebraic curves of $P_2(\mathbb{C})$. Mo…
The pentablock is a Hartogs domain over the symmetrized bidisc. The domain is a bounded inhomogeneous pseudoconvex domain, and does not have a $\mathcal{C}^{1}$ boundary. Recently, Agler-Lykova-Young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock, and Kosiński completely described the group of holomorphic automorphisms of the pentablock. The purpose of this paper is to prove that any proper holomorphic self-mapping of the pentablock must be an automorphism.
We establish basic results of complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds (such as algebras of Bohr's holomorphic almost periodic functions on tube domains or algebras of all fibrewise bounded holomorphic functions arising, e.g., in the corona problem for $H^\infty$). In particular, in this context we obtain results on holomorphic extension from complex submanifolds, properties of divisors, corona type theorems, holomorphic analogues of the Peter-Weyl approximation theorem, Hartogs type theorems, characterizations of uniqueness sets,…
We develop complex function theory within certain algebras of holomorphic functions on coverings of Stein manifolds. This, in particular, includes the results on holomorphic extension from complex submanifolds, corona type theorems, properties of divisors, holomorphic analogs of the Peter-Weyl approximation theorem, Hartogs type theorems, characterization of uniqueness sets. The model examples of these algebras are: (1) Bohr's algebra of holomorphic almost periodic functions on tube domains; (2) algebra of all fibrewise bounded holomorphic functions (e.g., arising in the corona problem for $H^…
The Fock-Bargmann-Hartogs domain $D_{n,m}(μ)$ ($μ>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-μ\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. Recently, Tu-Wang obtained the rigidity result that proper holomorphic self-mappings of $D_{n,m}(μ)$ are automorphisms for $m\geq 2$, and found a counter-example to show that the rigidity result isn't true for $D_{n,1}(μ)$. In this article, we obtain a classification of proper holomorphic mappings between $D_{n,1}(μ)$ and $D_{N,1}(μ)$ with $N<2n$.
We describe a part of the recent developments in the theory of separately holomorphic mappings between complex analytic spaces. Our description focuses on works using the technique of holomorphic discs.
We first exhibit counterexamples to some open questions related to a theorem of Sakai. Then we establish an extension theorem of Sakai type for separately holomorphic/meromorphic functions.
This paper has been withdrawn by the author(s) and included into the new version of "An extension theorem for separately holomorphic functions with singularities", math.CV/0104089.
We prove in this note a result on extension of meromorphic mappings, which can be considered as a direct generalisation of the Hartogs extension theorem for holomorphic functions. Namely: THEOREM. Every meromorphic mapping $f:H_n^q(r)\to Y$, where $Y$ is a $q$ - -complete complex space, extends to a meromorphic mapping from $Δ^{n+q}$ to $Y$. Here $H_n^q(r):=Δ^n\times (Δ^q\setminus \barΔ_r^q)\cup Δ_r^n\times Δ^q$ is a "q-concave" Hartogs figure in $C^{n+q}$. Remark that in the case $q=1$, i.e. when $Y$ is Stein, the statement of the Theorem is exactly the Theorem of Hartogs.
We extend the theory of separately holomorphic mappings between complex analytic spaces. Our method is based on Poletsky theory of discs, Rosay Theorem on holomorphic discs and our recent joint-work with Pflug on cross theorems in dimension 1. It also relies on our new technique of conformal mappings and a generalization of Siciak's relative extremal function. Our approach illustrates the unified character: ``From local informations to global extensions". Moreover, it avoids systematically the use of the classical method of doubly orthogonal bases of Bergman type.
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…
We consider the family $\mathcal P$ of $n$-tuples $P$ consisting of polynomials $P_1, \ldots, P_n$ with nonnegative coefficients which satisfy $\partial_i P_j(0) = δ_{i, j},$ $i, j=1, \ldots, n.$ With any such $P,$ we associate a Reinhardt domain $\triangle^{\!n}_{_P}$ that we will call the generalized Hartogs triangle. We are particularly interested in the choices $P_a = (P_{1, a}, \ldots, P_{n, a}),$ $a \geq 0,$ where $P_{j, a}(z) = z_j + a \prod_{k=1}^n z_k,~ j=1, \ldots, n.$ The generalized Hartogs triangle associated with $P_a$ is given by \begin{equation} \triangle^{\!n}_a = \Big\{z \in …
We establish extension theorems for separately holomorphic mappings defined on sets of the form W\setminus M with values in a complex analytic space which possesses the Hartogs extension property. Here W is a 2-fold cross of arbitrary complex manifolds and M is a set of singularities which is locally pluripolar (resp. thin) in fibers.
The purpose of this paper is to generalize in a geometric setting theorems of Severi, Brown and Bochner about analytic continuation of real analytic functions which are holomorphic or harmonic with respect to one of its variables. We prove in particular that if N is a real analytic levi-flat annulus in an open set of R^{n}\timesC^{2}, then one can find X\subsetR^{n}\timesC^{2} such that X\cupN is a levi-flat real analytic subset and X fills N in the sense that the boundary of the integration current of X is a prescribed smooth submanifold of N foliated by real curves. Moreover, real analytic f…
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 $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $Ω\subset {\mathbb{C}}^n \times {\mathbb{R}}$, $n \geq 2$, be a bounded domain with a connected real-analytic boundary such that $\partial Ω$ has only nondegenerate CR singularities. We prove that if $f \…
The Umehara algebra is studied with motivation on the problem of the non-existence of common complex submanifolds. In this paper, we prove some new results in Umehara algebra and obtain some applications. In particular, if a complex manifolds admits a holomorphic polynomial isometric immersion to one indefinite complex space form, then it cannot admits a holomorphic isometric immersion to another indefinite complex space form of different type. Other consequences include the non-existence of the common complex submanifolds for indefinite complex projective space or hyperbolic space and a compl…
Let $Ω\subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial Ω$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial Ω\to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial Ω$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\barΩ$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial Ω$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\barΩ$ in $…
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…
Let $\mathbb{B}^2$ denote the open unit ball in $\mathbb{C}^2$, and let $p\in \mathbb{C}^2\setminus\overline{\mathbb{B}^2}$. We prove that if $f$ is an analytic function on the sphere $\partial\mathbb{B}^2$ that extends holomorphically in each variable separately and along each complex line through $p$, then $f$ is the trace of a holomorphic function in the ball.
A subbundle of a Hermitian vector bundle $(E, h)$ can be metrically and differentiably defined by the orthogonal projection onto this subbundle. A weakly holomorphic subbundle of a Hermitian holomorphic bundle is, by definition, an orthogonal projection $π$ lying in the Sobolev space $L^2_1$ of $L^2$ sections with $L^2$ first order derivatives in the sense of distributions, which satisfies furthermore $(\mathrm{Id}-π)\circ D''π=0$. We give a new simple proof of the fact that a weakly holomorphic subbundle of $(E, h)$ defines a coherent subsheaf of ${\cal O}(E),$ that is a holomorphic subbundle…
