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Minus log boundary distance is plurisubharmonic on a half-space

Example

For the half-space

$H = {z \in C^{m} : Re z_{1} > 0},$

one has

$δ_{H} (z) = Re z_{1}, - \log δ_{H} (z) = - \log (Re z_{1}),$

and this function is plurisubharmonic on $H$ .

Facts & Assumptions

Given: The half-space $H = {Re z_{1} > 0}$ .

[L1]

Hartogs pseudoconvexity is defined through the function $- \log δ_{H}$ (Plurisubharmonic exhaustions and Hartogs pseudoconvexity).

[L2]

A $C^{2}$ function is plurisubharmonic exactly when its Levi form is semipositive (The C^2 Levi criterion for plurisubharmonicity).

Verification

technique · direct

1.1L1givenalgebra

The equal-radius polydisc about $z$ stays in the half-space exactly while its first-coordinate radius is smaller than $Re z_{1}$ , so $δ_{H} (z) = Re z_{1}$ . The function $u (z) = - \log (Re z_{1})$ is $C^{2}$ on $H$ and satisfies $\frac{\partial^{2} u}{\partial z_{1} \partial {\overline{z}}_{1}} (z) = \frac{1}{4 (Re z_{1})^{2}}, \frac{\partial^{2} u}{\partial z_{j} \partial {\overline{z}}_{k}} (z) = 0 for (j, k) \neq (1, 1) .$

2.1L2step 1.1∎

Hence the Levi form of $u$ is $L_{u} (z; v) = \frac{∣ v_{1} ∣^{2}}{4 (Re z_{1})^{2}} \geq 0.$ By [L2], $u = - \log δ_{H}$ is plurisubharmonic on $H$ .

Depends on

Used by

Nothing in the library uses this result yet.

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, Theorem 8 (standard reference, not scraped)