MiS Preprint Repository

A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is mean-convex and $\mathrm{\Gamma }\subset \partial \mathrm{\Omega }$ is a null-homotopic (in $\mathrm{\Omega }$) Jordan curve, then there exists an embedded minimal disk $\mathrm{\Sigma }\subset \overline{\mathrm{\Omega }}$ with boundary $\mathrm{\Gamma }$. Does a mean-convex set $\mathrm{\Omega }$ contain all minimal disks with boundary on $\partial \mathrm{\Omega }$? Does it contain the solutions of Plateau's problem? We answer this question negatively and characterize the least barrier enclosing all the minimal hypersurfaces with boundary on a given set.