The exterior algebra Λ kRn and area minimization

Abstract

The structure of the exterior algebra Λ k R n is studied in low dimensions, and consequences are drawn for k-dimensional area-minimizing surfaces in R n. For a general form ø∈Λ 2 R 4, Section 3 gives explicit formulas for the comass ‖ø‖ and the face of the Grassmannian exposed by ø. Section 4 classifies the faces of the Grassmannian of the 3-planes in R 6 and hence the associated geometries of area-minimizing surfaces (there are four types). Section 5 establishes an equality involving the comass norm in low dimension and draws implications on when the Cartesian product of area-minimizing surfaces is area-minimizing. New examples of area-minimizing integral currents with singularities follow.