2D quantum gravity on compact Riemann surfaces with non-conformal matter

We study the gravitational action induced by coupling two-dimensional non-conformal, massive matter to gravity on a compact Riemann surface. We express this gravitational action in terms of finite and well-de fined quantities for any value of the mass. A small-mass expansion gives back the Liouville action in the massless limit, the Mabuchi and Aubin-Yau actions to first order, as well as an in finite series of higher-order contributions written in terms of purely geometric quantities.