  The behavior of holographic CFTs is constrained by the existence of a bulk
dual geometry. For example, in (2+1)-dimensional holographic CFTs living on a
static spacetime with compact spatial slices, the vacuum energy must be
nonpositive, certain averaged energy densities must be nonpositive, and the
spectrum of scalar operators is bounded from below by the Ricci scalar of the
CFT geometry. Are these results special to holographic CFTs? Here we show that
for perturbations about appropriate backgrounds, they are in fact universal to
all CFTs, as they follow from the universal behavior of two- and three-point
correlators of known operators. In the case of vacuum energy, we extend away
from the perturbative regime and make global statements about its negativity
properties on the space of spatial geometries. Finally, we comment on the
implications for dynamics which are dissipative and driven by such a vacuum
energy and we remark on similar results for the behavior of the Euclidean
partition function on deformations of flat space or the round sphere.
