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Topological characterization of Stein manifolds of dimension $$>2$$. (English) Zbl 0699.58002
In this paper is given a topological characterization of Stein manifolds of dimension $$>2.$$
The main result is the following: Let X be a 2n-dimensional smooth manifold, $$n>2$$, with an almost complex structure J and assume that there exists a proper Morse function $$\phi$$ : $$X\to {\mathbb{R}}$$ such that the indexes of all its critical points are $$\leq n$$. Then there exists a complex structure $$\tilde J$$ on X such that $$(X,\tilde J)$$ is Stein. In fact the author shows that the new structure $$\tilde J$$ can be chosen such that J is homotopic to $$\tilde J$$ and $$\phi$$ is $$\tilde J$$-convex. By a well-known result of Grauert $$(X,\tilde J)$$is Stein.
Reviewer: M.Colţoiu

##### MSC:
 58A05 Differentiable manifolds, foundations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 32E10 Stein spaces, Stein manifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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