We show that any bounded balanced domain of holomorphy is an $L{²}_h$-domain of holomorphy.

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 $h:M\to M^{\text{'}}$ between two globally minimal real analytic hypersurfaces in ${ℂ}^n$ ($n\ge 2$) is real analytic at every point...

We study the extension problem of holomorphic maps $\sigma :H\to X$ of a Hartogs domain $H$ with values in a complex manifold $X$. For compact Kähler manifolds as well as various non-Kähler manifolds, the maximal domain ${\Omega }_{\sigma }$ of extension for $\sigma$ over $\Delta$ is contained in a subdomain of $\Delta$. For such manifolds, we define, in this paper, an invariant Hex${}_n\left(X\right)$ using the Hausdorff dimensions of the singular sets of $\sigma$’s and study its properties to deduce informations on the complex structure of $X$.

To a pair of a Lie group $G$ and an open elliptic convex cone $W$ in its Lie algebra one associates a complex semigroup $S=G\mathrm{Exp}\left(iW\right)$ which permits an action of $G\times G$ by biholomorphic mappings. In the case where $W$ is a vector space $S$ is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain $D\subseteq S$ is Stein is and only if it is of the form $G\mathrm{Exp}\left(D_h\right)$, with $Dh\subseteq iW$ convex, that each holomorphic function on $D$ extends to the smallest biinvariant Stein domain containing $D$,...

Let $M=G/H$ be a real symmetric space and $𝔤=𝔥+𝔮$ the corresponding decomposition of the Lie algebra. To each open $H$-invariant domain $D_{𝔮}\subseteq i𝔮$ consisting of real ad-diagonalizable elements, we associate a complex manifold $\Xi \left(D_{𝔮}\right)$ which is a curved analog of a tube domain with base $D_{𝔮}$, and we have a natural action of $G$ by holomorphic mappings. We show that $\Xi \left(D_{𝔮}\right)$ is a Stein manifold if and only if $D_{𝔮}$ is convex, that the envelope of holomorphy is schlicht and that $G$-invariant plurisubharmonic functions correspond to convex $H$-invariant...

If E is a closed subset of locally finite Hausdorff (2n-2)-measure on an n-dimensional complex manifold Ω and all the points of E are nonremovable for a meromorphic mapping of Ω E into a compact Kähler manifold, then E is a pure (n-1)-dimensional complex analytic subset of Ω.

We study some properties of the maximal ideal space of the bounded holomorphic functions in several variables. Two examples of bounded balanced domains are introduced, both having non-trivial maximal ideals.