On the conditional existence of foliations by CMC and Willmore type half-spheres (Q6592223)

Let \(\Omega \subset \mathbb{R}^3\) be a bounded smooth domain and \(S\) the boundary of \(\Omega\).\N\textit{R. Alessandroni} and \textit{E. Kuwert} [Calc. Var. Partial Differ. Equ. 55, No. 2, Paper No. 24, 29 p. (2016; Zbl 1344.49066)] constructed and studied area-constrained Willmore half-spheres with small areas meeting \(S\) orthogonally along the boundaries. In particular, they showed that such half-spheres concentrate at critical points of the mean curvature functional \(H^S\) of \(S\). \N\NIn the present paper, the author gives a criterion for existence of a foliation by such half-spheres near a nondegenerate critical point of \(H^S\) (Theorem 1.1).\N\N\textit{G. Belletini} and \textit{G. Fusco} [J. Differ. Equations 157, No. 1, 206--246 (1999; Zbl 0968.53044)] constructed and studied CMC half-spheres meeting \(S\) orthogonally along the boundaries and giving small enclosed volumes. \NIn the present paper, the author gives a criterion analogous to the criterion in Theorem 1.1 for such half-spheres (Theorem 1.2).