  Third sound measurements of superfluid $^4$He thin films adsorbed on 10 nm
diameter multiwall carbon nanotubes are used to probe the superfluid onset
temperature as a function of the film thickness, and to study the temperature
dependence of the film compressibility. The nanotubes provide a highly ordered
carbon surface, with layer-by-layer growth of the adsorbed film as shown by
oscillation peaks in the third sound velocity at the completion of the third,
fourth, and fifth atomic layers, arising from oscillations in the
compressibility. In temperature sweeps the third sound velocity at very low
temperatures is found to be linear with temperature, but oscillating between
positive and negative slope depending on the film thickness. Analysis shows
that this can be attributed to a linearly decreasing compressibility of the
film with temperature that appears to hold even near zero temperature. The
superfluid onset temperature is found to be linear in the film thickness, as
predicted by the Kosterlitz-Thouless theory, but the slope is anomalous, a
factor of three smaller than the predicted universal value.
