Calibrated Geometry - History of Calibrations (Geometric Measure Theory, Chapter 6, Section 6.5, Frank Morgan)

History of Calibrations (Geometric Measure Theory, Chapter 6, Section 6.5, Frank Morgan)

The original example of complex analytic varieties was implicit in Wilhelm Wirtinger (1936), explicit for complex analytic submanifolds in Georges de Rham (1957), and applied to singular complex varieties in the context of rectifiable currents by Herbert Federer (1965). Marcel Berger (1970) was the first to extract the underlying principle and apply it to other examples such as quaternionic varieties,followed by Dao (1977). The term calibration was coined in the landmark paper of Harvey and Lawson, which discovered rich new calibrated geometries of “special Lagrangian,” “associative,” and “Cayley” varieties. The method has grown in power and applications. Surveys appear in Morgan . Mackenzie and Lawlor use calibrations in the proof (Nance; Lawlor ) of the angle conjecture on when a pair of m-planes in Rn is area minimizing. The “vanishing calibrations” of Lawlor actually provide sufficient differential-geometric conditions for area minimization, a classification of all area-minimizing cones over products of m spheres, examples of nonorientable area-minimizing cones, and singularities stable under perturbations. The “paired calibrations” of Lawlor and Morgan and of Brakke prove new examples of soap films in R3, in R4, and above. Other developments include Murdoch’s “twisted calibrations” of nonorientable surfaces, Le’s “relative calibrations” of stable surfaces, and Pontryagin calibrations on Grassmannians (Gluck, Mackenzie, and Morgan). Lawlor has developed a related theory for proving minimization by slicing. Lawlor and Morgan show, for example, that three minimal surfaces meeting at 120 degrees minimize area locally.