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A boundary cross theorem for separately holomorphic functions

Peter Pflug, Viêt-Anh Nguyên (2004)

Annales Polonici Mathematici

Let D ⊂ ℂⁿ and G ⊂ 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 result to N-fold...

A differential geometric characterization of invariant domains of holomorphy

Gregor Fels (1995)

Annales de l'institut Fourier

Let G = K ℂ be a complex reductive group. We give a description both of domains Ω ⊂ G and plurisubharmonic functions, which are invariant by the compact group, K , acting on G by (right) translation. This is done in terms of curvature of the associated Riemannian symmetric space M : = G / K . Such an invariant domain Ω with a smooth boundary is Stein if and only if the corresponding domain Ω M ⊂ M is geodesically convex and the sectional curvature of its boundary S : = ∂ Ω M fulfills the condition K S ( E ) ≥ K M ( E ) + k ( E , n ) . The term k ( E , n ) is explicitly computable...

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

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 × Y ) ∪ ( X × B ) → Z extends to a holomorphic mapping f ^ on W ^ : = ( z , w ) ∈ X × Y : ω ˜ ( z , A , X ) + ω ˜ ( w , B , Y ) < 1 such that f ^ = f ...

A microlocal version of Cartan-Grauert's theorem

I. V. Maresin, A. G. Sergeev (1998)

Annales Polonici Mathematici

Tuboids are tube-like domains which have a totally real edge and look asymptotically near the edge as a local tube over a convex cone. For such domains we state an analogue of Cartan’s theorem on the holomorphic convexity of totally real domains in ℝ n ⊂ ℂ n .

A note on composition operators on spaces of real analytic functions

Paweł Domański, Michał Goliński, Michael Langenbruch (2012)

Annales Polonici Mathematici

We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semiproper map φ such that the associated composition operator is not open onto its image.

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