  We discuss several aspects of the proposed correspondence between quantum
gravity on de Sitter spaces and Euclidean conformal field theories. The central
charge appearing in the asymptotic symmetry algebra of three-dimensional de
Sitter space is derived both from the conformal anomaly and the transformation
law of the CFT stress tensor when going from dS_3 in planar coordinates to dS_3
with cosmological horizon. The two-point correlator for CFT operators coupling
to bulk scalars is obtained in static coordinates, corresponding to a CFT on a
cylinder. Correlation functions are also computed for CFTs on two-dimensional
hyperbolic space. We furthermore determine the energy momentum tensor and the
Casimir energy of the conformal field theory dual to the Schwarzschild-de
Sitter solution in five dimensions. Requiring the pressure to be positive
yields an upper bound for the black hole mass, given by the mass of the Nariai
solution. Beyond that bound, which is similar to the one found by Strominger
requiring the conformal weights of CFT operators to be real, one encounters
naked singularities.
