A Sufficient Criterion for a Cone to Be Area-Minimizing

This book presents a systematic algorithm for proving that certain cones are area-minimizing. The problem of determining what shapes of singularities can occur has stimulated much research since the 1960s when it was discovered that area-minimizing surfaces can have essential singularities. One landmark was the work of Reese Harvey and Blaine Lawson on calibrations, which helped to motivate Frank Morgan's correct conjecture in 1981 concerning which pairs of planes are area-minimizing. This conjecture was a catalyst for a wave of fruitful ideas, including the method presented in this book. The algorithm the author describes consists of examining a first order ordinary differential equation based on the curvature and dimension of the cone and ensuring that certain line segments normal to the cone do not intersect. The method is novel in the wide variety of cones to which it can be systematically applied. Many new examples are provided, including the completion of the classification of minimizing cones over products of two or more spheres (in a few cases, such as products of spheres, the criterion is necessary and sufficient). Though the book is written primarily for those in geometric measure theory and differential geometry, much of it is accessible to mathematicians in other areas, as well as to graduate students and advanced undergraduates interested in area-minimizing surfaces.

A minimization test for cones; Calibrations; The differential equation; Cones for which the criterion is necessary as well as sufficient; Examples of area-minimizing cones; Some perturbation results; Open questions.