On the Conditional Existence of Foliations by CMC and Willmore Type Half-Spheres

TL;DR

This paper investigates the conditions under which families of small CMC and Willmore half-spheres form foliations near boundary critical points, revealing that such foliations depend on boundary geometry and differ from Riemannian cases.

Contribution

It establishes a criterion linking boundary geometry to the existence of foliations by CMC and Willmore half-spheres, highlighting a conditional phenomenon absent in Riemannian settings.

Findings

Foliations are not guaranteed near boundary critical points.

Boundary geometry determines the existence of such foliations.

Symmetry considerations lead to unconditional foliations in Riemannian cases.

Abstract

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'.