Levi form

Summary

The Levi form is a Hermitian form built from the mixed second derivatives $\partial^2 \rho / \partial z_j \partial \bar z_k$ of a real function $\rho$ on a complex domain. For the boundary of a domain in $\C^n$ one restricts it to the complex tangent directions of the boundary. Its sign decides the convexity notions of several complex variables: pseudoconvexity, strong pseudoconvexity and plurisubharmonicity.