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Hartogs pseudoconvexity yields a continuous plurisubharmonic exhaustion

Statement

Let $Ω \subseteq C^{m}$ be a domain. If $Ω$ is Hartogs pseudoconvex, that is, if $- \log δ_{Ω}$ is plurisubharmonic on $Ω$ , then $Ω$ admits a continuous plurisubharmonic exhaustion function.

Facts & Assumptions

Given: A domain $Ω \subseteq C^{m}$ .

[L1]

Hartogs pseudoconvexity means exactly that $- \log δ_{Ω}$ is plurisubharmonic, while a continuous plurisubharmonic exhaustion is defined by compact sublevel sets (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[L2]

Finite maxima preserve plurisubharmonicity (Basic stability operations for plurisubharmonic functions).

[L3]

The squared norm has positive Levi form, so it is plurisubharmonic (The Levi form and strict plurisubharmonicity, The C^2 Levi criterion for plurisubharmonicity).

Proof

technique · direct

1.1

Assume that $- \log δ_{Ω}$ is plurisubharmonic. The function $q (z) = ∣ z_{1} ∣^{2} + \dots + ∣ z_{m} ∣^{2}$ is plurisubharmonic by [L3]. Since $δ_{Ω}$ is a distance-to-the-complement function for the sup norm, it is continuous on $Ω$ , so $- \log δ_{Ω}$ is continuous as well. Therefore [L2] makes $u (z) : = \max {- \log δ_{Ω} (z), q (z)}$ a continuous plurisubharmonic function.

L2L3given

2.1

As $z$ approaches $\partial Ω$ , the term $- \log δ_{Ω} (z)$ tends to $+ \infty$ , and as $∣ z ∣ \to \infty$ inside $Ω$ , the term $q (z)$ tends to $+ \infty$ . Hence the sublevel sets of $u$ are compact in $Ω$ , so $u$ is a continuous plurisubharmonic exhaustion.

L1step 1.1given∎

Depends on

Used by

Hartogs pseudoconvexity implies Levi pseudoconvexity for C² domains Theorem

Dependency tree · two levels

8 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, Theorem 2.5.6 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, Theorem 8 (standard reference, not scraped)