Convex symplectic manifolds.

*(English)*Zbl 0742.53010
Several complex variables and complex geometry, Proc. Summer Res. Inst., Santa Cruz/CA (USA) 1989, Proc. Symp. Pure Math. 52, Part 2, 135-162 (1991).

[For the entire collection see Zbl 0732.00008.]

A symplectic structure on an even dimensional smooth manifold \(V^{2n}\) is a closed nonsingular exterior differential form \(\omega\) of degree 2, and as is well-known, a theorem of Darboux says that any symplectic manifold \((V,\omega)\) is locally isomorphic to \((R^{2n},\sum^ n_{i,j=1}dx^ i\wedge dx^ j)\). A \((V,\omega)\) is said to be convex complete, if it admits a vector field \(\partial\) such that (1) \(\partial\) is contracting, i.e. \(\partial\omega=\lambda\omega\) for some \(\lambda < 0\). (2) \(\partial\) is integrable, i.e. \(\partial\) generates a one- parameter group of homeomorphisms \(A_ t: V\to V\), (\(t\in R\)). (3) \(\partial\) is strictly compact, i.e. \(V\) is covered by open subsets \(U_ i\), \(i=1,2,\dots \), such that \(A_ t\) maps \(U_ i\) onto a relatively compact subset in \(U_ i\) for all \(t>0\) and \(i=1,2,\dots .\)

Its basic examples are furnished by the usual cotangent bundles and Stein manifolds equipped with appropriate Kählerian metrics. Because of this, the main goal of this paper may be said to confirm that such convex complete manifolds are similar in many respects to compact symplectic manifolds and their behavior is governed by the laws of topology rather than by those of geometry, as the conditions (2) and (3) show.

A symplectic structure on an even dimensional smooth manifold \(V^{2n}\) is a closed nonsingular exterior differential form \(\omega\) of degree 2, and as is well-known, a theorem of Darboux says that any symplectic manifold \((V,\omega)\) is locally isomorphic to \((R^{2n},\sum^ n_{i,j=1}dx^ i\wedge dx^ j)\). A \((V,\omega)\) is said to be convex complete, if it admits a vector field \(\partial\) such that (1) \(\partial\) is contracting, i.e. \(\partial\omega=\lambda\omega\) for some \(\lambda < 0\). (2) \(\partial\) is integrable, i.e. \(\partial\) generates a one- parameter group of homeomorphisms \(A_ t: V\to V\), (\(t\in R\)). (3) \(\partial\) is strictly compact, i.e. \(V\) is covered by open subsets \(U_ i\), \(i=1,2,\dots \), such that \(A_ t\) maps \(U_ i\) onto a relatively compact subset in \(U_ i\) for all \(t>0\) and \(i=1,2,\dots .\)

Its basic examples are furnished by the usual cotangent bundles and Stein manifolds equipped with appropriate Kählerian metrics. Because of this, the main goal of this paper may be said to confirm that such convex complete manifolds are similar in many respects to compact symplectic manifolds and their behavior is governed by the laws of topology rather than by those of geometry, as the conditions (2) and (3) show.

Reviewer: T.Okubo (Victoria)