  We study the gravitational action induced by coupling two-dimensional
non-conformal, massive matter to gravity on a Riemann surface with boundaries.
A small-mass expansion gives back the Liouville action in the massless limit,
while the first-order mass correction allows us to identify what should be the
appropriate generalization of the Mabuchi action on a Riemann surface with
boundaries. We provide a detailed study for the example of the cylinder.
Contrary to the case of manifolds without boundary, we find that the
gravitational Lagrangian explicitly depends on the space-point, via the
geodesic distances to the boundaries, as well as on the modular parameter of
the cylinder, through an elliptic theta-function.
