  Many of the existing results for closed Hamiltonian G-manifolds are based on
the analysis of the corresponding Hamiltonian functions using Morse-Bott
techniques. In general such methods fail for non-compact manifolds or for
manifolds with boundary.
  In this article, we consider circle actions only on symplectic manifolds that
have (convex) contact type boundary. In this situation we show that many of the
key ideas of Morse-Bott theory still hold, allowing us to generalize several
results from the closed setting.
  Among these, we show that in our situation any symplectic group action is
always Hamiltonian, we show several results about the topology of the
symplectic manifold and in particular about the connectedness of its boundary.
We also show that after attaching cylindrical ends, a level set of the
Hamiltonian of a circle action is either empty or connected.
  We concentrate mostly on circle actions, but we believe that with our methods
many of the classical results can be generalized from closed symplectic
manifolds to symplectic manifolds with contact type boundary.
