Mean-convex sets and minimal barriers

Emanuele Spadaro

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Submission date: 06. Jan. 2012
Pages: 25
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Abstract:
A mean-convex set can be regarded as a barrier for the construction of minimal surfaces. Namely, if Ω ⊂ ℝ³ is mean-convex and Γ ⊂ ∂Ω is a null-homotopic (in Ω) Jordan curve, then there exists an embedded minimal disk Σ ⊂Ω with boundary Γ. Does a mean-convex set Ω contain all minimal disks with boundary on ∂Ω? 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.