A CONVEX DECOMPOSITION THEOREM FOR
S. AKBULUT AND R. MATVEYEV
Abstract. In this article we show that every smooth closed oriented four-
manifold admits a decomposition into two submanifolds along common bound-
ary. Each of these submanifolds is a complex manifold with pseudo-convex
boundary. This imply, in particular, that every smooth closed simply-connected
four-manifold is a Stein domain in the the complement of a certain contractible
Exact manifold with pseudo-convex boundary (PC manifold, for short) is a com-
pact complex manifold X, which admits strictly pluri-subharmonic Morse function
, such that set of maximum points of coincides with the boundary X. We
prefer the term PC manifold, since combination of words "compact Stein manifold"
is likely to precipitate heart palpitations in some mathematicians.
Such manifold admits a symplectic structure = i
2 and serves as an analogue
of closed symplectic manifold. Boundary X of PC manifold X inherits a contact
structure , which, in this case, is a distribution of maximal complex subspaces in