We discuss some implications of the gravitational dressing of the reno
rmalization group for conformal field theories perturbed by relevant o
perators. The renormalization group flows are defined with respect to
the dilatation operator associated with the J(0)((0)) mode of the SL(2
, R) affine algebra. We discuss the possibility of passing under the c
= 25 barrier along renormalization group flows in some models.