---
tag: 6VF47Q35YA
kind: statement
role: theorem
name: Embedding theorem for Stein manifolds
msc: [32Q40, 32H02, 32Q28]
uses: [HMCD0JBFST, TM528Q96ZD, 3MQTWKBBBT]
about: [HMCD0JBFST]
sources:
  - url: https://en.wikipedia.org/wiki/Stein_manifold
---

## Statement

Let $X$ be a [[Stein manifold|HMCD0JBFST]] of complex dimension $n$.
Then there is a [[proper|TM528Q96ZD]] injective [[holomorphic map|3MQTWKBBBT]] $F \colon X \to \C^{2n+1}$ whose differential is injective at every point.
So $X$ is biholomorphic to a closed complex submanifold of $\C^{2n+1}$.

## Remarks

The theorem is due to Remmert, Bishop and Narasimhan.
The proof starts from the many global holomorphic functions that Cartan's theorems A and B provide on a Stein manifold.
With them one builds a proper injective holomorphic map into some $\C^N$ with injective differential.
Then one lowers $N$ by generic linear projections, as in the proof of the Whitney embedding theorem.
The converse holds: every closed complex submanifold of $\C^N$ is Stein, because the coordinate functions separate points and give compact holomorphically convex hulls.
So the Stein manifolds are exactly the closed complex submanifolds of complex Euclidean spaces.
Example: a noncompact Riemann surface, which is Stein, embeds properly into $\C^3$.
The dimension $2n + 1$ is not optimal.
For $n \ge 2$, Eliashberg and Gromov, with a final step by Schürmann, showed that $\C^N$ with $N = \floor{3n/2} + 1$ suffices, and this value is optimal.

## Explanations

### L1 human

The theorem says that every Stein manifold can be drawn inside some $\C^N$ by holomorphic functions, without self-intersections.
Going off to infinity in the manifold then means going off to infinity in $\C^N$.
So the abstract definition of a Stein manifold, by conditions on its holomorphic functions, describes the same objects as the concrete closed submanifolds of $\C^N$.
It is the complex-analytic analogue of the Whitney embedding theorem for smooth manifolds.

### L2 human

Embeddings reduce questions about Stein manifolds to questions about submanifolds of $\C^N$.
For example, $\abs{F}^2$ is a smooth strictly plurisubharmonic exhaustion function, which gives the easy half of Grauert's characterization of Stein manifolds.
Whether every open Riemann surface embeds properly in $\C^2$ is an open problem.