  We apply a novel background independent regularization scheme, the
$N$-cutoffs, to self-consistently quantize scalar and metric fluctuations on
the maximally symmetric but non-compact hyperbolic space. For quantum matter
fields on a classical background or full Quantum Einstein Gravity (regarded
here as an effective field theory) treated in the background field formalism,
the $N$-cutoff is an ultraviolet regularization of the fields' mode content
that is independent of the background hyperbolic space metric. For each $N >
0$, the regularized system backreacts on the geometry to dynamically determine
the self-consistent background metric. The limit in which the regularization is
removed then automatically yields the 'physically correct' spacetime on which
the resulting quantum field theory lives. When self-consistently quantized with
the $N$-cutoff, we find that without any fine-tuning of parameters, the vacuum
fluctuations of scalar and (linearized) graviton fields do not lead to the
usual cosmological constant problem of a curvature singularity. Instead, the
presence of increasingly many field modes tends to reduce the negative
curvature of hyperbolic space, leading to vanishing values in the limit of
removing the cutoff.
