Department of MathematicsIn Morse theory, a function with non-degenerate singularities can be used to decompose a smooth manifold into handles. Similarly, we can build symplectic handlebodies by gluing symplectic handles, so a natural question is: when does a Morse function induce a decomposition into symplectic handles? The answer is, the function has to be compatible with a Liouville vector field on M. Whenever such a vector field and such a function exist, M is said to be Weinstein.
On a different note, one can consider a version of Morse functions for complex geometry, called Lefschetz fibrations. As it turns out, there is a very strong link between Lefschetz fibrations and Weinstein manifolds!