  We study half-spheres with small radii sitting on the boundary of a smooth
bounded domain while meeting it orthogonally. Even though it is known that
there exist families of CMC and Willmore type half-spheres near a nondegenerate
critical point p of the domains boundaries mean curvature, it is unknown in
both cases whether these provide a foliation of any deleted neighborhood of p.
We prove that this is not guaranteed and establish a criterion in terms of the
boundaries geometry that ensures or prevents the respective surfaces from
providing such a foliation. This perhaps surprising phenomenon of conditional
foliations is absent in the closely related Riemannian setting, where a
foliation is guaranteed. We show how this unconditional foliation arises from
symmetry considerations and how these fail to apply to the 'domain-setting'.
