Abstract
In the present article, the author proves two generalizations of his “finiteness-result” (I.H.P. Anal. Non-lineaire, 2006, accepted) which states for any extreme simple closed polygon \(\Gamma \subset {\mathbb{R}}^3\) that every immersed, stable disc-type minimal surface spanning Γ is an isolated point of the set of all disc-type minimal surfaces spanning Γ w.r.t. the C 0-topology. First, it is proved that this statement holds true for any simple closed polygon in \({\mathbb{R}}^3\) , provided it bounds only minimal surfaces without boundary branch points. Also requiring that the interior angles at the vertices of such a polygon Γ have to be different from \(\frac{\pi }{2}\) the author proves the existence of some neighborhood O of Γ in \({\mathbb{R}}^3\) and of some integer \(\beta \) , depending only on Γ, such that the number of immersed, stable disc-type minimal surfaces spanning any simple closed polygon contained in O, with the same number of vertices as Γ, is bounded by \(\beta\) .
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Jakob, R. Local boundedness of the number of solutions of Plateau’s problem for polygonal boundary curves. Ann Glob Anal Geom 33, 231–244 (2008). https://doi.org/10.1007/s10455-007-9082-8
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DOI: https://doi.org/10.1007/s10455-007-9082-8