Soap films bounded by non-closed curves
type: Item
instance of: scholarly article
title: Soap films bounded by non-closed curves
zbMATH Open document ID: 0959.58018
DOI: 10.1007/BF02921642
author: Jordan Drachman
author: Brian White
published in: The Journal of Geometric Analysis
publication date: 2001-01-16T00:00:00Z
review text: Usually when discussing soap films on a curve formed from wire, one imagines that the curve is closed. Nonetheless, it was noted by \textit{F. J. Almgren jun.} in 1966 [Plateau's problem. An invitation to varifold geometry, W. A. Benjamin, New York (1966; Zbl 0165.13201)] that soap films can form on non-closed curves. To model such a situation mathematically requires a very general context for the surfaces under consideration such as the following used by the authors: The set \(A\subset{\mathbb R}^{3}\) is said to be area minimizing with respect to another set \(\gamma\) if whenever \(B\) is a ball disjoint from \(\gamma\) and \(f:B\to B\) is a Lipschitz map that leaves points of \(\partial B\) fixed, it holds that \({\mathcal H}^{2}(A\cap B)\leq{\mathcal H}^{2}(f(A\cap B)).\) The main theorem of this paper uses the notion of the total curvature of a \(C^{1}\) curve in \({\mathbb R}^{3}.\) It is simpler to define the total curvature for a \(C^{2}\) curve, but an extension to the \(C^{1}\) case was given by \textit{J. W. Milnor} using polygonal approximations [Ann. Math. (2) 52, 248-257 (1950; Zbl 0037.38904)]. The main theorem of the paper states that any non-closed simple \(C^{1}\) curve in \({\mathbb R}^{3}\) for which there is set of positive two dimensional Hausdorff measure that is area minimizing with respect to the curve must have total curvature strictly greater than \(2\pi.\) Moreover, any total curvature greater than \(2\pi\) is realized by some non-closed simple curve for which there exists a set of positive two dimensional Hausdorff measure that is area minimizing with respect to that curve. To show that no curve of total curvature less than or equal to \(2\pi\) bounds an area minimizing set, the authors employ a lemma of Milnor to show that one of the endpoints is advantageously positioned below a half-space containing the rest of the curve. They then repeatedly apply the maximum principle of \textit{B. Solomon} and \textit{B. White} [Indiana Univ. Math…
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 1353668
zbMATH Keywords: soap films
zbMATH Keywords: minimal surfaces
zbMATH Keywords: knotted curves
reviewed by: Harold Parks
MaRDI profile type: Publication
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cites work: Q5822904
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cites work: The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces
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description: scientific article; zbMATH DE number 1353668