  In the low-density surface region of superfluid $^4$He, the atoms are far
apart and collisions can be ignored. The only effect of the interactions is
from the long-range attractive Hartree potential produced by the distant
high-density bulk liquid. As a result, at $T=0$, all the atoms occupy the same
single-particle state in the low-density tail. Striking numerical evidence for
this 100\% surface BEC was given by Pandharipande and coworkers in 1988. We
derive a generalized Gross-Pitaevskii equation for the inhomogeneous condensate
wave function $\Phi(z)$ in the low-density region valid at all temperatures.
The overall amplitude of $\Phi(z)$ is fixed by the bulk liquid, which ensures
that it vanishes everywhere at the bulk transition temperature.
