Stein manifold is a generalization of the concept of the domain of holomorphy to manifolds. Furthermore, Stein manifolds are the generalizations of Riemann surfaces in higher dimensions. Every noncompact Riemann surface is a Stein manifold by a theorem of Behnke and Stein. Note that every domain of holomorphy in is a Stein manifold. It is not hard to see that every closed complex submanifold of a Stein manifold is Stein.
Theorem (Remmert, Narasimhan, Bishop).
If is a Stein manifold of dimension . There exists a proper (http://planetmath.org/ProperMap) holomorphic embedding of into .
Note that no compact complex manifold can be Stein since compact complex manifolds have no holomorphic functions. On the other hand, every compact complex manifold is holomorphically convex.
- 1 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 2 Steven G. Krantz. , AMS Chelsea Publishing, Providence, Rhode Island, 1992.
|Date of creation||2013-03-22 15:04:37|
|Last modified on||2013-03-22 15:04:37|
|Last modified by||jirka (4157)|