Free boundary minimal hypersurfaces with least area (Q6598525)

Let \(M\) be a compact orientable Riemannian manifold with boundary \(\partial M\). Free boundary minimal hypersurfaces in \(M\) have zero mean curvature and their boundaries meet \(\partial M\) orthogonally. Suppose \(3\leq \dim M \leq 7\). Let \(\mathcal{O}\) (resp. \(\mathcal{U}\)) be the set of embedded compact orientable (resp. non-orientable) free boundary minimal hypersurfaces in \(M\). Then building on [\textit{L. Mazet} and \textit{H. Rosenberg}, J. Differ. Geom. 106, No. 2, 283--316 (2017; Zbl 1404.53078)], the authors show that there exists an embedded compact free boundary minimal hypersurface in \(M\) which attains the infimum of the sum of \(\{\mathrm{Area}\,(\Sigma ) \ | \ \Sigma \in \mathcal{O} \}\) and \(\{ 2\,\mathrm{Area}\,(\Sigma ) \ | \ \Sigma \in \mathcal{U} \}\).