ONE-LOOP EFFECTIVE POTENTIAL IN 2-D DILATON GRAVITY ON HYPERBOLIC PLANE

The one-loop effective potential in 2-D dilaton gravity in conformal g auge on the topologically non-trivial plane R x S1 and on the hyperbol ic plane H-2/GAMMA is calculated. For arbitrary choice of the tree sca lar potential it is shown that the one-loop effective potential explic itly depends on the reference metric (through the dependence on the ra dius of the torus or the radius of H-2/GAMMA). This phenomenon is abse nt only for some special choice of the tree scalar potential correspon ding to the Liouville potential and leading to one-loop uv finite theo ry. The effective equations are also discussed.