Effective Lagrangian for self-interacting scalar field theories in curved spacetime

We consider a self-interacting scalar field theory in a slowly varying gravitational background field. Using zeta-function regularization and heat-kernel techniques, we derive the one-loop effective Lagrangian up to second order in the variation of the background field and up to quadratic terms in the curvature tensors. Specializing to different spacetimes of physical interest, the influence of the curvature on the phase transition is considered.