Soap films bounded by non-closed curves
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. J. 38, No. 3, 683-691 (1989; Zbl 0711.49059)]. The existence of sets that are area minimizing with respect to non-closed curves is obtained by using curves that are geometrically knotted. Heuristically, a cylinder is placed around the knotted curve and a minimizing sequence of homotopic cylinders is considered. The fact that the curve is knotted bounds the areas away from zero and results of \textit{F. J. Almgren jun.} [Symp. Math. 14, 413-428 (1974; Zbl 0315.49018)] show the sequence has an area minimizing limit.
scientific article; zbMATH DE number 1353668
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