Stein manifold

The Stein's manifold is an object from the higher dimensional function theory. Named this was after the mathematician Karl Stein. Stein's A manifold is a special complex manifold. It is the natural definition set of holomorphic functions, because it ensures that there are enough holomorphic functions; So in addition to the constant functions more holomorphic functions exist.

Definition

With one call to the set of holomorphic functions. A complex manifold of dimension is called Stein's diversity, if

• Is holomorphically convex, ie

• Is holomorphically separable, ie for two different points in, there is a holomorphic function with

Examples

• Every domain of holomorphy is a Stein's manifold.
• Be a submanifold of a Stein's manifold. If is complete, then a Stein's diversity again.
• A Riemann surface is a Stein's manifold if and only if it is not compact.

Embedding theorem

Every real -dimensional differentiable manifold can be embedded in the after embedding theorem of Whitney. This result is false for complex manifolds in general. Compact complex manifolds one can not embed in the example. However, Stein's manifolds can always embed. The following theorem was proved by Reinhold Remmert and Errett Bishop.

Be a Stein's manifold of dimension, then there exists a holomorphic map which is injective and actually.

In that case you can embed any -dimensional manifold in the Stein's. For this, you can even embed in the. Here is the floor function which rounds up the value to the nearest integer.