Levi pseudoconvex

Let Gn be a domain (http://planetmath.org/Domain2) (open connected subset) with C2 boundary, that is the boundary is locally the graph of a twice continuously differentiable function. Let ρ:n be a defining function of G, that is ρ is a twice continuously differentiable function such that gradρ(z)0 for zG and G={znρ(z)<0} (such a function always exists).


Let pG (boundary of G). We call the space of vectors w=(w1,,wn)n such that


the space of holomorphic tangent vectors at p and denote it Tp1,0(G).

Tp1,0(G) is an n-1 dimensional complex vector space and is a subspacePlanetmathPlanetmath of the complexified real tangent space (http://planetmath.org/TangentSpace), that is Tp(G).

Note that when n=1 then the complex tangent space contains just the zero vectorMathworldPlanetmath.


The point pG is called Levi pseudoconvex (or just pseudoconvex) if


for all wTp1,0(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 Tp1,0(G) and not on all of n.


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 C2 boundary is a domain of holomorphy if and only if G is Levi pseudoconvex.


  • 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