Superforms, supercurrents, minimal manifolds and Riemannian geometry

Abstract

AbstractSupercurrents, as introduced by Lagerberg, were mainly motivated as a way to study tropical varieties. Here we will associate a supercurrent to any smooth submanifold of $${\mathbb {R}}^n$$Rn. Positive supercurrents resemble positive currents in complex analysis, but depend on a choice of scalar product on $${\mathbb {R}}^n$$Rn and reflect the induced Riemannian structure on the submanifold. In this way we can use techniques from complex analysis to study real submanifolds. We illustrate the idea by giving area estimates of minimal manifolds and a monotonicity property of the mean curvature flow. We also use the formalism to give a relatively short proof of Weyl’s tube formula.