---
tag: Z7RVXBA2KB
kind: concept
name: Strongly pseudoconvex domain
msc: [32T15, 32T05]
specializes: [PETEQQCS0B]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoconvexity
---

## Summary

A strongly pseudoconvex domain is a domain in $\C^n$ with $C^2$ boundary whose [[Levi form|80MTNXWW80]] is positive definite at every boundary point.
It is also called a strictly pseudoconvex domain.
It is the complex analogue of a strictly convex domain.
In this class the analysis of [[several complex variables|BH57D2FBNR]] works best.
The $\bar\partial$-equation has good estimates, and every boundary point is locally convex after a holomorphic change of coordinates.

## Explanations

### L1 human

A strongly pseudoconvex domain bends outwards strictly in every complex direction along its boundary.
The model is the unit ball in $\C^n$.
Small smooth perturbations of the ball stay strongly pseudoconvex, so there are very many such domains.
The bidisc is not one, because its boundary contains flat complex discs, such as $\set{(1, z_2) : \abs{z_2} < 1}$.