  There is a great deal of interest in attributing the high critical
temperatures of the cuprates to either the proximity of the Fermi level to a
van Hove singularity or to structure of the superconducting pairing potential
in momentum space far from the Fermi surface. We examine these ideas by
calculating the critical temperature Tc for model Einstein-phonon- and
spin-fluctuation-mediated superconductors within both the standard,
Fermi-surface-restricted Eliashberg theory and the exact mean field theory,
which accounts for the full momentum structure of the pairing potential and the
energy dependence of the density of states. By using two models of
spin-fluctuation-mediated pairing in the cuprates, we demonstrate that our
results are independent of the details of the dynamical susceptibility, which
is taken to be the pairing potential. We also compare these two models against
available neutron scattering data, since these data provide the most direct
constraints on the susceptibility. We conclude that the van Hove singularity
does not drastically alter Tc from its value when the density of states is
constant and that the effect of momentum structure is significant but secondary
in importance to that of the energy dependence in the density of states.
