  We present a complete calculation of the structure of liquid $^4$He confined
to a concave nanoscopic wedge, as a function of the opening angle of the walls.
This is achieved within a finite-range density functional formalism. The
results here presented, restricted to alkali metal substrates, illustrate the
change in meniscus shape from rather broad to narrow wedges on weak and strong
alkali adsorbers, and relate this change to the wetting behavior of helium on
the corresponding planar substrate. As the wedge angle is varied, we find a
sequence of stable states that, in the case of cesium, undergo one filling and
one emptying transition at large and small openings, respectively. A
computationally unambiguous criterion to determine the contact angle of $^4$He
on cesium is also proposed.
