---
tag: ZKG8D1S566
kind: definition
name: Levi pseudoconvex domain
msc: [32T05]
defines: PETEQQCS0B
uses: [80MTNXWW80]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoconvexity
---

## Definition

Let $\Omega \subseteq \C^n$ be a domain with $C^2$ boundary, and let $\rho$ be a defining function of $\Omega$.
For $p \in \partial\Omega$, let $H_p$ be the complex tangent space of $\partial\Omega$ at $p$.
The domain $\Omega$ is *Levi pseudoconvex* if the [[Levi form|80MTNXWW80]] of $\partial\Omega$ is positive semidefinite at every boundary point:
$$\sum_{j,k=1}^{n} \frac{\partial^2 \rho}{\partial z_j \partial \bar z_k}(p)\, w_j \bar w_k \ge 0 \quad \text{for all } p \in \partial\Omega \text{ and all } w \in H_p.$$

## Remarks

The condition does not depend on the choice of $\rho$, because another defining function multiplies the Levi form by a positive number.
Example: every convex domain with $C^2$ boundary is Levi pseudoconvex.
Indeed, for a convex domain the real Hessian of $\rho$ at $p$ is positive semidefinite on the real tangent space of $\partial\Omega$ at $p$.
For $w \in H_p$, both $w$ and $i w$ lie in this tangent space, and $L_\rho(p; w)$ is one quarter of the sum of the real Hessian at $w$ and at $i w$.
Non-example: the domain $\set{z \in \C^2 : 1 < \abs{z_1}^2 + \abs{z_2}^2 < 4}$ fails at the inner sphere.
There $\rho(z) = 1 - \abs{z}^2$ is a defining function, and $L_\rho(p; w) = -\abs{w}^2 < 0$ for $w \in H_p$, $w \ne 0$.
When $n = 1$ the space $H_p$ is $\set{0}$, so every planar domain with $C^2$ boundary is Levi pseudoconvex.