Steinian manifold

The Stein manifold is an object from the higher-dimensional function theory . This was named after the mathematician Karl Stein . A Stein manifold is a special complex manifold . It is the natural set of definitions of holomorphic functions, because it is ensured that there are enough holomorphic functions; so besides the constant functions there are further holomorphic functions.

definition

With call to the set of holomorphic functions on . A complex manifold of dimension is called Stein's manifold, if ${\displaystyle {\mathcal {O}}(M)}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle n}$

• ${\displaystyle M}$ is holomorphic convex , that is

${\displaystyle {\hat {K}}=\{z\in M;|f(z)|\leq \sup _{K}|f|\ \mathrm {f{\ddot {u}}r\,jedes} \ f\in {\mathcal {O}}(M)\}}$

is a compact subset of for each compact subset . ${\displaystyle M}$ ${\displaystyle K\subset M}$

• ${\displaystyle M}$is holomorphically separable , i.e. for two different points and in , there is a holomorphic function with ${\displaystyle z_{1}}$ ${\displaystyle z_{2}}$ ${\displaystyle M}$ ${\displaystyle f\in {\mathcal {O}}(M)}$

${\displaystyle f(z_{1})\neq f(z_{2})}$.

Examples

• Every holomorphic domain is a Steinian manifold.
• Let be a submanifold of a Stein's manifold. If it is closed , then there is again a Stein manifold. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$
• A Riemann surface is a Stein manifold if and only if it is not compact . ${\displaystyle M}$ ${\displaystyle M}$

Embedding set

Every real -dimensional differentiable manifold can be embedded in the according to Whitney's embedding theorem . This result is generally wrong for complex manifolds. For example, compact, complex manifolds of positive dimensions cannot be embedded in the. However, Stein's manifolds can always be embedded. The following theorem has been proven by Reinhold Remmert and Errett Bishop . ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{2n}}$ ${\displaystyle \mathbb {C} ^{N}}$

Let the dimension be a Stein manifold , then there exists a holomorphic mapping which is injective and real . ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle f\colon M\to \mathbb {C} ^{2n+1}}$

In that case one can embed every -dimensional Steinian manifold in the . For you can even embed them in the . Here is the Gaussian bracket , which rounds the value up to the nearest whole number. ${\displaystyle n\geq 2}$ ${\displaystyle n}$ ${\displaystyle \mathbb {C} ^{2n}}$ ${\displaystyle n>6}$ ${\displaystyle \mathbb {C} ^{2n-\left[(n-2)/3\right]}}$ ${\displaystyle [.]}$

literature

• Klaus Fritzsche, Hans Grauert : From Holomorphic Functions to Complex Manifolds (= Graduate Texts in Mathematics 213). Springer, New York NY et al. 2002, ISBN 0-387-95395-7 .
• Lars Hörmander : An Introduction to Complex Analysis in Several Variables (= North-Holland Mathematical Library 7). 2nd revised edition. North-Holland et al., Amsterdam et al. 1973, ISBN 0-7204-2450-X .