On the conditional existence of foliations by CMC and Willmore type half-spheres

Abstract

We study half-spheres with small radii [Formula: see text] 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 [Formula: see text] of the domains boundaries mean curvature, it is unknown in both cases whether these provide a foliation of any deleted neighborhood of [Formula: see text]. 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”.