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Good candidates for least area soap films. (English) Zbl 1256.49054

Summary: Soap films are presented as potential global area minimizers subject to a topological constraint. Experimentally, this constraint is the shape of the soapy water in a soap film experiment. In this context, soap films which are probable area minimizers for rectangular \(n\)-prisms are described. By allowing area minimizers which arise as deformations of higher genus surfaces, we are able to discover previously unknown soap films spanning rectangular \(n\)-prisms with large aspect ratios and \(n \geq 5\). For \(n = 3, 4, 5\), we show that the central film contracts to a point as the aspect ratio of the prism increases. We also prove that the area of the central hexagon for a soap film spanning a tall 6-prism approaches zero like (height)\(^{ - 4}\) as the height approaches infinity, provided we fix the length of the hexagon base. Finally, we prove that, if the aspect ratio is large enough, the soap film produced experimentally spanning a 4-prism has films which look planar but in reality are non-planar.

MSC:

49Q05 Minimal surfaces and optimization
51M04 Elementary problems in Euclidean geometries
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