Superforms, supercurrents, minimal manifolds and Riemannian geometry
arXiv:1805.00379
Abstract
Supercurrents, 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 . Positive supercurrents resemble positive currents in complex analysis, but depend on a choice of scalar product on 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 monotinicity property of the mean curvature flow. We also illustrate the idea by a relatively short proof of Weyl's tube formula.
An incorrect proof in section 8 (on mean curvature flow) has been replaced; the main results are unchanged. Some additional explanations have also been added. 25 pages