Let be a domain (http://planetmath.org/Domain2) (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).
Let (boundary of ). We call the space of vectors such that
the space of holomorphic tangent vectors at and denote it .
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 .
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.
- 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.
|Date of creation||2013-03-22 14:30:37|
|Last modified on||2013-03-22 14:30:37|
|Last modified by||jirka (4157)|
|Defines||strongly Levi pseudoconvex|
|Defines||weakly Levi pseudoconvex|
|Defines||holomorphic tangent vector|