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∂̅-cohomology and geometry of the boundary of pseudoconvex domains

Takeo Ohsawa (2007)

Annales Polonici Mathematici

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...

Hyperconvexity of non-smooth pseudoconvex domains

Xu Wang (2014)

Annales Polonici Mathematici

We show that a bounded pseudoconvex domain D ⊂ ℂⁿ is hyperconvex if its boundary ∂D can be written locally as a complex continuous family of log-Lipschitz curves. We also prove that the graph of a holomorphic motion of a bounded regular domain Ω ⊂ ℂ is hyperconvex provided every component of ∂Ω contains at least two points. Furthermore, we show that hyperconvexity is a Hölder-homeomorphic invariant for planar domains.

Remarks on pluripolar hulls

Le Mau Hai, Nguyen Quang Dieu, Tang Van Long (2004)

Annales Polonici Mathematici

The aim of the paper is to establish some results on pluripolar hulls and to define pluripolar hulls of certain graphs.

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