Topological characterization of Stein manifolds of dimension \(>2\). (English) Zbl 0699.58002

Eliashberg, Yakov

Int. J. Math. 1, No. 1, 29-46 (1990).

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
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

topological characterization of Stein manifolds; smooth manifold; almost complex structure