  We introduce a class of Weinstein domains which are sublevel sets of flexible
Weinstein manifolds but are not themselves flexible. These manifolds exhibit
rather subtle behavior with respect to both holomorphic curve invariants and
symplectic flexibility. We construct a large class of examples and prove that
every flexible Weinstein manifold can be Weinstein homotoped to have a
nonflexible sublevel set.

Symplectic Geometry