DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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Plurisubharmonic exhaustions and Hartogs pseudoconvexity

Definition

Let $Ω \subseteq C^{m}$ be a domain.

A function $u : Ω \to R$ is a continuous plurisubharmonic exhaustion when $u$ is continuous, plurisubharmonic, and every sublevel set

${z \in Ω : u (z) \leq c}$

is compact in $Ω$ for every real number $c$ .

The domain $Ω$ is Hartogs pseudoconvex when the function

$z ⟼ - \log δ_{Ω} (z)$

is plurisubharmonic on $Ω$ , where $δ_{Ω}$ is the equal-radius polydisc boundary function of The equal-radius polydisc boundary function.

When $Ω = C^{m}$ , one has $δ_{Ω} \equiv + \infty$ ; in this sole case, the displayed boundary function is by convention the constant function $0$ . Thus the whole space is Hartogs pseudoconvex.

Depends on

Used by

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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)