In this paper we introduce a new method for proving area-minimization which we call "paired calibrations." We begin with the simplest application, the cone over the tetrahedron, which appears in soap films. We then discuss immiscible fluid interfaces, crystal surfaces, and one-dimensional networks minimizing other norms.
Summary.We complete the proof of the angle conjecture on when a pair of oriented m-planes is area-minimizing. The nonzero sum (oriented union) P~ + P2 is area-minimizing if and only if the characterizing angles between them satisfy the inequality
We use a new approach that we call unification to prove that standard weighted double bubbles in n-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight) double bubble theorem [HHS], [HMRR], [R].As part of the proof we introduce a striking new symmetry argument for showing that a minimizer must be a surface of revolution.
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