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We study the gravitational action induced by coupling two-dimensionalnon-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 theappropriate generalization of the Mabuchi action on a Riemann surface withboundaries. We provide a detailed study for the example of the cylinder.Contrary to the case of manifolds without boundary, we find that thegravitational Lagrangian explicitly depends on the space-point, via thegeodesic distances to the boundaries, as well as on the modular parameter ofthe cylinder, through an elliptic theta-function.

2D gravitational Mabuchi action on Riemann surfaces with boundaries