### A Characterization of Complex Manifolds Biholomorphic to a Circular Domain.

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Let $G={K}^{\u2102}$ be a complex reductive group. We give a description both of domains $\Omega \subset G$ 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 $\Omega $ with a smooth boundary is Stein if and only if the corresponding domain ${\Omega}_{M}\subset M$ is geodesically convex and the sectional curvature of its boundary $S:=\partial {\Omega}_{M}$ fulfills the condition ${K}^{S}\left(E\right)\ge {K}^{M}\left(E\right)+k(E,n)$. The term $k(E,n)$ is explicitly computable...

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 ${\mathbb{R}}^{n}\subset {\u2102}^{n}$.

Let X be a Riemann domain over ${\u2102}^{k}\times {\u2102}^{\ell}$. If X is a domain of holomorphy with respect to a family ℱ ⊂(X), then there exists a pluripolar set $P\subset {\u2102}^{k}$ such that every slice ${X}_{a}$ of X with a∉ P is a region of holomorphy with respect to the family ${f|}_{{X}_{a}}:f\in \mathcal{F}$.

In 1958, H. Grauert proved: If D is a strongly pseudoconvex domain in a complex manifold, then D is holomorphically convex. In contrast, various cases occur if the Levi form of the boundary of D is everywhere zero, i.e. if ∂D is Levi flat. A review is given of the results on the domains with Levi flat boundaries in recent decades. Related results on the domains with divisorial boundaries and generically strongly pseudoconvex domains are also presented. As for the methods, it is explained how Hartogs...

Let ${N}_{\alpha}$,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and ${Q}_{\beta}$ are Möbius invariant, but ${N}_{\alpha}$ is not. We characterize, in function-theoretic terms, when the composition operator ${C}_{\varphi}f=f\u25e6\varphi $ induced by an analytic self-map ϕ of the unit disk defines an operator ${C}_{\varphi}:{N}_{\alpha}\to B$, $B\to {Q}_{\beta}$, ${N}_{\alpha}\to {Q}_{\beta}$ which is bounded resp. compact.

We prove that a Cousin-I open set D of an irreducible projective surface X is locally Stein at every boundary point which lies in ${X}_{reg}$. In particular, Cousin-I proper open sets of ℙ² are Stein. We also study K-envelopes of holomorphy of K-complete spaces.

If Ω is a domain of holomorphy in Cn, having a compact topological closure into another domain of holomorphy U ⊂ Cn such that (Ω,U) is a Runge pair, we construct a function F holomorphic in Ω which is singular at every boundary point of Ω and such that F is in Lp(Ω), for any p ∈ (0, +∞).