Symplectic circle actions on manifolds with contact type boundary

Many existing results for closed, Hamiltonian $G$-manifolds rely on analyzing the corresponding Hamiltonian functions with Morse-Bott techniques. In general however, such methods fail for non-compact manifolds or manifolds with boundary. In this article, we consider circle actions on symplectic manifolds with (convex) contact type boundary. We show that many key ideas of Morse-Bott theory still hold in this situation, thereby allowing us to generalize several results from the closed setting. For example, we prove that a symplectic group action is always Hamiltonian, that the contact type boundary of a Hamiltonian $G$-manifold is always connected except possibly for $G = \mathbb{S}^1$ and that several other results about the topology of the symplectic manifold hold. We also show that after attaching cylindrical ends, a level set of the Hamiltonian of a circle action is either empty or connected. Although we focus primarily on circle actions, it is clear that many other classical results about symplectic group actions can be generalized with our methods, extending them from closed symplectic manifolds to those with a contact type boundary.