Let be a domain (open connected subset) with boundary, that is the boundary is locally the graph of a twice continuously differentiable function. Let be a defining function of , that is is a twice continuously differentiable function such that for and (such a function always exists).

Definition 1   Let (boundary of ). We call the space of vectors such that

the space of holomorphic tangent vectors at and denote it .

is an dimensional complex vector space and is a subspace of the complexified real tangent space, that is .

Note that when then the complex tangent space contains just the zero vector.

Definition 2   The point is called Levi pseudoconvex (or just pseudoconvex) if

for all . 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 dimensional space, so the expression above may be confusing as it only acts on and not on all of .

Definition 3   The domain is called Levi pseudoconvex if every boundary point is Levi pseudoconvex. Similarly 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 with boundary is a domain of holomorphy if and only if is Levi pseudoconvex.

Bibliography

1

M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.

2

Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.