Abstract
In this article, the author proves that a simple closed polygon \({\Gamma\subset\mathbb{R}^3}\) can bound only finitely many immersed minimal surfaces of disc-type if it meets the following two requirements: firstly it has to bound only minimal surfaces without boundary branch points, and secondly its total curvature, i.e. the sum of the exterior angles \({\{\eta_l\}}\) at its N + 3 vertices, has to be smaller than 6π.
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Jakob, R. Finiteness of the number of solutions of Plateau’s problem for polygonal boundary curves II. Ann Glob Anal Geom 36, 19–35 (2009). https://doi.org/10.1007/s10455-008-9146-4
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DOI: https://doi.org/10.1007/s10455-008-9146-4