  We investigate gravitational correlation functions in a curved background
with the help of nonperturbative renormalization group methods. Beta functions
for eleven couplings are derived, two of which correspond to running gauge
parameters. A unique ultraviolet fixed point is found, suitable for a UV
completion in the sense of Asymptotic Safety. To arrive at a well-behaved flow
in a curved background, the regularization must be chosen carefully. We provide
two admissible choices to solve this issue in the present approximation. We
further demonstrate by an explicit calculation that the Landau limit is a fixed
point also for quantum gravity, and additionally show that in this limit, the
gauge parameter $\beta$ does not flow.
