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MiS Preprint
2/2012
Mean-convex sets and minimal barriers
Emanuele Spadaro
Abstract
A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if $\Omega \subset \mathbb{R}^3$ is mean-convex and $\Gamma \subset \partial\Omega$ is a null-homotopic (in $\Omega$) Jordan curve, then there exists an embedded minimal disk $\Sigma \subset \bar\Omega$ with boundary $\Gamma$. Does a mean-convex set $\Omega$ contain all minimal disks with boundary on $\partial\Omega$? 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.
