  The effects of a heat current and gravity in liquid 4He near the superfluid
transition are investigated for temperatures above and below T_lambda. We
present a renormalization-group calculation based on model F for the Green's
function in a self-consistent approximation which in quantum many-particle
theory is known as the Hartree approximation. The approach can handle a zero
average order parameter above and below T_lambda and includes effects of
vortices. We calculate the thermal conductivity and the specific heat for all
temperatures T and heat currents Q in the critical regime. Furthermore, we
calculate the temperature profile. Below T_lambda we find a second correlation
length which describes the dephasing of the order parameter field due to
vortices. We find dissipation and mutual friction of the superfluid-normal
fluid counterflow and calculate the Gorter-Mellink coefficient A. We compare
our theoretical results with recent experiments.
