A Sufficient Criterion For A Cone To Be Area-minimizing (memoirs Of The American Mathematical Society)

One of the fundamental objects of study in geometric measure theory is an "area-minimizing surface." A compact, k-dimensional surface (with boundary) is called area-minimizing if no other surface with the same boundary has less surface area. An area-minimizing surface can have singularities. The main purpose of this paper is to investigate some of the shapes that such singularities can have. A key concept in this study is that of an area-minimizing cone. We present a general method for proving that a cone with an isolated singularity is area-minimizing. The calculation involves the curvature (second fundamental form) and a sort of "embedding radius" of the normal bundle to the cone. We can also prove that certain cones are not area-minimizing. Using this method, we complete the classification of minimizing cones over products of spheres. We also give other examples, including the first known unorientable minimizing cones. The method also lends itself to perturbation arguments. We show that certain surfaces are area-minimizing in a small neighborhood of an isolated singularity