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The unit ball is Levi pseudoconvex

Example

For $m \geq 1$ , use the canonical coordinates indexed by $j < m$ of Complex $m$ -space and its real coordinate dictionary. The unit ball

$B = {z \in C^{m} : \sum_{j < m} ∣ z_{j} ∣^{2} < 1}$

is Levi pseudoconvex.

Facts & Assumptions

Given: $m \geq 1$ , canonical indices $j < m$ , and the function $ρ (z) = \sum_{j < m} ∣ z_{j} ∣^{2} - 1$ of the unit ball.

[L1]

Levi pseudoconvexity is tested by the Levi form of a defining function on complex tangent vectors (Levi pseudoconvex domains).

[L2]

The Levi form is $L_{ρ} (z; v) = \sum_{j, k < m} \frac{\partial^{2} ρ}{\partial z_{j} \partial {\overline{z}}_{k}} (z) v_{j} \overline{v_{k}}$ (The Levi form and strict plurisubharmonicity, reindexed to the canonical coordinates $j, k < m$ ).

[L3]

Canonical complex coordinates are $z_{j} = x_{j} + i y_{j}$ for $j < m$ , and $\sum_{j < m} ∣ z_{j} ∣^{2} = \sum_{j < m} (x_{j}^{2} + y_{j}^{2})$ is the squared Euclidean norm in the real coordinate identification. (Complex $m$ -space and its real coordinate dictionary)

Verification

technique · direct

1.1L2givenalgebra

For $j, k < m$ and $ρ (z) = \sum_{j < m} ∣ z_{j} ∣^{2} - 1$ , one has $\frac{\partial^{2} ρ}{\partial z_{j} \partial {\overline{z}}_{k}} = δ_{j k},$ so [L2] gives $L_{ρ} (z; v) = \sum_{j < m} ∣ v_{j} ∣^{2} \geq 0$ for every $z$ and every $v \in C^{m}$ .

1.2L3givenalgebra

The function $ρ$ is a real polynomial, and $B = {ρ < 0}$ . At a boundary point $p$ its real coordinates satisfy $\sum_{j < m} (x_{j}^{2} + y_{j}^{2}) = 1$ by [L3], so some real coordinate $X$ is nonzero and $\partial_{X} ρ (p) = 2 X (p) \neq 0$ . Locally the equation $ρ = 0$ solves that coordinate as $X = sgn (X (p)) \sqrt{1 - \sum Y^{2}}$ , where $Y$ ranges over the other real coordinates; the radicand is positive near $p$ . Thus the boundary is smooth and $ρ$ is a $C^{2}$ local defining function with nonzero differential.

2.1L1step 1.1step 1.2∎

In particular the Levi form is nonnegative on every complex tangent vector at every boundary point of the unit ball. By [L1], the unit ball is Levi pseudoconvex.

Depends on

Used by

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Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, §2.3 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.3.1 (standard reference, not scraped)