Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms
In the classification of soap-film singularities \textit{J. Taylor} [Ann. Math., II. Ser. 103, 489-539 (1976; Zbl 0335.49032)] proved only by the process of elimination that the cone over the edges of the regular tetrahedron minimizes area among surfaces separating the four faces. The purpose of the present paper is to give a direct proof which applies to regular simplices in all dimensions. After an introductory section, the second section provides examples of cones which minimize total interface energy, as for immiscible fluids. These cones serve as models for general singular structure. The third section provides examples of cones which minimize hypersurface energies given by general norms \(\Phi_{ij}\) on the space of normal vectors, as in the surface energy of crystals. The case when all of the norms \(\Phi_{ij}\) are equal is of primary interest. Examples include a cone over a triangular prism (Proposition 3.5). Finally, the section four classifies the singularities in \(\Phi\)- minimizing networks in \(\mathbb{R}^n\) and establishes that in \(\mathbb{R}^n\) at most \(n+ 1\) segments meet at a point (Theorem 4.5).
scientific article; zbMATH DE number 759234