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Levi pseudoconvexity does not depend on the defining function

Statement

Let $Ω \subseteq C^{m}$ have $C^{2}$ boundary, let $p \in \partial Ω$ , and let $ρ$ and $\tilde{ρ}$ be two $C^{2}$ defining functions near $p$ . Then on complex tangent vectors at $p$ , the Levi forms differ by a positive scalar factor. In particular, the sign condition in the definition of Levi pseudoconvexity is independent of the defining function.

Facts & Assumptions

Given: A boundary point $p \in \partial Ω$ and two $C^{2}$ defining functions $ρ$ and $\tilde{ρ}$ near $p$ .

[L1]

Levi pseudoconvexity is stated in terms of the Levi form on complex tangent vectors of a defining function (Levi pseudoconvex domains).

Proof

technique · direct

1.1

Because $ρ$ and $\tilde{ρ}$ vanish on the same $C^{2}$ hypersurface, have nonzero differentials there, and define the same negative side, one has $\tilde{ρ} = h ρ$ near $p$ for a positive $C^{1}$ function $h$ .

L1given

2.1

Let $v$ be a complex tangent vector at $p$ , so $\partial ρ (p) \cdot v = 0$ . The second-order expansion of $h ρ$ at $p$ uses only first derivatives of $h$ because $ρ (p) = 0$ ; every mixed product term contains $\partial ρ (p) \cdot v$ or its conjugate and therefore vanishes on $v$ . Consequently $L_{\tilde{ρ}} (p; v) = h (p) L_{ρ} (p; v) .$ Since $h (p) > 0$ , the two Levi forms have the same sign on complex tangent vectors.

step 1.1algebra∎

Depends on

Levi pseudoconvex domains

Used by

Hartogs pseudoconvexity implies Levi pseudoconvexity for C² domains Theorem

Dependency tree · two levels

2 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, §2.3 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, §3.3.1 (standard reference, not scraped)