The intrinsic metric of constant mean curvature surfaces and minimal hypersurfaces with free boundary

Ricci-Curbastro established necessary and sufficient conditions for a Riemannian metric on a surface to be the first fundamental form of a minimal immersion of that surface into the Euclidean space. We revisit certain developments arising from his theorem, and propose new versions of these results in the context of the theory of constant mean curvature surfaces in three-dimensional space forms that meet umbilical surfaces orthogonally along their boundary components. Higher dimensional generalisations, inspired by a theorem of do Carmo and Dajczer, are discussed as well.