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Plurisubharmonic exhaustions and Hartogs pseudoconvexity
Definition
Let be a domain.
A function is a continuous plurisubharmonic exhaustion when is continuous, plurisubharmonic, and every sublevel set
is compact in for every real number .
The domain is Hartogs pseudoconvex when the function
is plurisubharmonic on , where is the equal-radius polydisc boundary function of The equal-radius polydisc boundary function.
When , one has ; in this sole case, the displayed boundary function is by convention the constant function . Thus the whole space is Hartogs pseudoconvex.
Depends on
Used by
- First Cousin problem on a pseudoconvex domain Corollary
- Positive-degree Dolbeault vanishing on pseudoconvex domains Corollary
- A Hartogs domain with a strictly plurisubharmonic exhaustion Example
- A strictly plurisubharmonic exhaustion of the convex unit ball Example
- An explicit ∂̄ solution with an L² estimate Example
- First Cousin gluing on the pseudoconvex domain ℂ Example
- Hörmander estimate with a Gaussian weight Example
- Minus log boundary distance is plurisubharmonic on a half-space Example
- A smooth psh exhaustion gives Hartogs pseudoconvexity on bounded domains Lemma
- Oka-Weil approximation on a domain of holomorphy (host-domain lemma) Lemma
- Positive smooth collars for strictly plurisubharmonic negative sets Lemma
- Smooth strict plurisubharmonic regularization of a psh exhaustion Lemma
- Behnke-Stein: increasing unions of pseudoconvex domains Theorem
- Domains of holomorphy are Hartogs pseudoconvex Theorem
- Hartogs pseudoconvexity implies Levi pseudoconvexity for C² domains Theorem
- Hartogs pseudoconvexity yields a continuous plurisubharmonic exhaustion Theorem
- Hörmander's weighted L2 existence theorem for the dbar equation Theorem
- Oka-Weil approximation on a pseudoconvex domain Theorem
- Smooth strictly plurisubharmonic exhaustion of a pseudoconvex domain Theorem
- The Levi problem: pseudoconvexity, domains of holomorphy, and holomorphic convexity Theorem
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Jiří Lebl, Tasty Bits of Several Complex Variables, §2.5 (standard reference, not scraped)
- Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.4 (standard reference, not scraped)