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In the low-density surface region of superfluid $^4$He, the atoms are farapart and collisions can be ignored. The only effect of the interactions isfrom the long-range attractive Hartree potential produced by the distanthigh-density bulk liquid. As a result, at $T=0$, all the atoms occupy the samesingle-particle state in the low-density tail. Striking numerical evidence forthis 100\% surface BEC was given by Pandharipande and coworkers in 1988. Wederive a generalized Gross-Pitaevskii equation for the inhomogeneous condensatewave 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 ensuresthat it vanishes everywhere at the bulk transition temperature.Comment: 6 pages, paper submitted to Low Temperature Conference (LT21),  Prague, Aug., 1996; to appear in proceeding

The surface region of superfluid4He as a dilute Bose-condensed gas