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The C^2 Levi criterion for plurisubharmonicity

Statement

Let $Ω \subseteq C^{m}$ be open and let $u \in C^{2} (Ω, R)$ . Then $u$ is plurisubharmonic on $Ω$ if and only if

$L_{u} (a; v) \geq 0 for every a \in Ω and every v \in C^{m} .$

Facts & Assumptions

Given: An open set $Ω \subseteq C^{m}$ and a function $u \in C^{2} (Ω, R)$ .

[L1]

Plurisubharmonicity is defined by subharmonicity of the restriction to every affine complex line (Plurisubharmonic functions).

[L2]

The Levi form is the Hermitian form built from the mixed $z_{j} {\overline{z}}_{k}$ second derivatives (The Levi form and strict plurisubharmonicity).

[L3]

A $C^{2}$ real-valued function of one complex variable is subharmonic exactly when its Laplacian is nonnegative (A C^2 function is subharmonic exactly when its Laplacian is nonnegative).

Proof

technique · direct

1.1L2L3givenalgebra

Fix $a \in Ω$ and $v \in C^{m}$ , and define $ϕ_{a, v} (λ) = a + λ v$ on a small disc about $0$ whose image lies in $Ω$ . By the chain rule, $\frac{\partial^{2}}{\partial λ \partial \overline{λ}} (u \circ ϕ_{a, v}) (0) = \sum_{j = 1}^{m} \sum_{k = 1}^{m} \frac{\partial^{2} u}{\partial z_{j} \partial {\overline{z}}_{k}} (a) v_{j} \overline{v_{k}} = L_{u} (a; v) .$ Since $Δ = 4 \partial_{λ} \partial_{\overline{λ}}$ in one complex variable, [L3] says that $u \circ ϕ_{a, v}$ is subharmonic exactly when $L_{u} (a; v) \geq 0$ .

2.1L1step 1.1∎

If $u$ is plurisubharmonic, then every line restriction is subharmonic by [L1], so step 1.1 gives $L_{u} (a; v) \geq 0$ for every $a$ and $v$ . Conversely, if the Levi form is semipositive everywhere, then step 1.1 and [L3] show that every affine-line restriction is subharmonic. Applying [L1] again, $u$ is plurisubharmonic on $Ω$ .

Depends on

Used by

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, §2.4 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.2.4 and §3.3.1 (standard reference, not scraped)