Size-minimizing rectifiable currents (Q1103186)

Traditionally in computing the area of a surface (rectifiable current) one counts multiplicities. Size is an alternative to area which does not count multiplicities. This paper gives some of the first existence results for size-minimizing rectifiable currents, for hypersurfaces with given external boundary. It presents what geometric measure theory can say about regularity, isoperimetric inequalities, and size bounds. Finally, it proves a generalization of Bernstein's theorem: a complete 2- dimensional size-minimizing surface S in \({\mathbb{R}}^ 3 \)is a plane, three half-planes meeting along a line at 120\(\circ\) angles, or the cone over the 1-skeleton of the regular tetrahedron.