# Levi pseudoconvex

Let $G\subset {\u2102}^{n}$ be a domain (http://planetmath.org/Domain2) (open connected subset) with ${C}^{2}$ boundary, that is the boundary is locally the graph of a twice continuously differentiable function. Let $\rho :{\u2102}^{n}\to \mathbb{R}$ be a defining function of $G$, that is $\rho $ is a twice continuously differentiable function such that $\mathrm{grad}\rho (z)\ne 0$ for $z\in \partial G$ and $$ (such a function always exists).

###### Definition.

Let $p\in \partial G$ (boundary of $G$). We call the space of vectors $w=({w}_{1},\mathrm{\dots},{w}_{n})\in {\u2102}^{n}$ such that

$$\sum _{k=1}^{n}\frac{\partial \rho}{\partial {z}_{k}}(p){w}_{k}=0,$$ |

the space of holomorphic tangent vectors at $p$ and denote it ${T}_{p}^{1,0}(\partial G)$.

${T}_{p}^{1,0}(\partial G)$ is an $n-1$ dimensional complex vector space
and is a subspace^{} of the complexified real tangent space (http://planetmath.org/TangentSpace), that is $\u2102{\otimes}_{\mathbb{R}}{T}_{p}(\partial G)$.

Note that when $n=1$ then the complex
tangent space contains just the zero vector^{}.

###### Definition.

The point $p\in \partial G$ is called Levi pseudoconvex (or just pseudoconvex) if

$$\sum _{j,k=1}^{n}\frac{{\partial}^{2}\rho}{\partial {z}_{j}\partial {\overline{z}}_{k}}(p){w}_{j}{\overline{w}}_{k}\ge 0,$$ |

for all $w\in {T}_{p}^{1,0}(\partial G)$. The point is called strongly Levi pseudoconvex (or just strongly pseudoconvex or also strictly pseudoconvex) if the inequality above is strict. The expression on the left is called the Levi form.

Note that if a point is not strongly Levi pseudoconvex then it is sometimes called a weakly Levi pseudoconvex point.

The Levi form really acts on an $n-1$ dimensional space, so the expression above may be confusing as it only acts on ${T}_{p}^{1,0}(\partial G)$ and not on all of ${\u2102}^{n}$.

###### Definition.

The domain $G$ is called Levi pseudoconvex if every boundary point is Levi pseudoconvex. Similarly $G$ is called strongly Levi pseudoconvex if every boundary point is strongly Levi pseudoconvex.

Note that in particular all convex domains are pseudoconvex.

It turns out that $G$ with ${C}^{2}$ boundary is a domain of holomorphy if and only if $G$ is Levi pseudoconvex.

## References

- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.

Title | Levi pseudoconvex |

Canonical name | LeviPseudoconvex |

Date of creation | 2013-03-22 14:30:37 |

Last modified on | 2013-03-22 14:30:37 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 10 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 32T15 |

Classification | msc 32T05 |

Related topic | DomainOfHolomorphy |

Related topic | Pseudoconvex |

Related topic | BiholomorphismsOfStronglyPseudoconvexDomainsExtendToTheBoundary |

Defines | Levi form |

Defines | strongly Levi pseudoconvex |

Defines | strongly pseudoconvex |

Defines | strictly pseudoconvex |

Defines | weakly pseudoconvex |

Defines | weakly Levi pseudoconvex |

Defines | holomorphic tangent vector |