How to construct metrics to control distributions of homologically mass-minimizing currents

TL;DR

This paper explores how to design metrics on Riemannian manifolds to ensure that most homologically mass-minimizing currents are simple linear combinations of smooth submanifolds, simplifying their regularity and distribution.

Contribution

It introduces methods for constructing metrics that make homologically mass-minimizing currents have regular, well-understood structures.

Findings

Metrics can be constructed to regularize mass-minimizing currents

Almost all such currents become linear combinations of smooth submanifolds

Enhances understanding of the geometric structure of mass-minimizing currents

Abstract

By Federer and Fleming there exist at least one mass-minimizing normal current in every real-valued homology class of a Riemannian manifold. However the regularity of the mass-minimizing currents and their distributions may generally be quite complicated. In this paper we shall study how to construct nice metrics so that (as functionals over smooth forms) almost all homologically mass-minimizing currents are just linear combinations over smooth submanifolds.