Stationary polyhedral varifolds minimize area

TL;DR

This paper discusses the area-minimizing properties of stationary polyhedral varifolds, showing they are minimal under certain deformations and relate to mass-minimizing flat chains, primarily providing an exposition of existing results.

Contribution

The paper offers an exposition of known results demonstrating that stationary polyhedral varifolds minimize area under specific conditions.

Findings

Stationary polyhedral varifolds cannot have their area decreased by certain Lipschitz deformations.

They are associated with mass-minimizing flat chains with coefficients in a metric abelian group.

Abstract

We prove that every stationary polyhedral varifold minimizes area in the following senses: (1) its area cannot be decreased by a one-to-one Lipschitz ambient deformation that coincides with the identity outside of a compact set, and (2) it is the varifold associated to a mass-minimizing flat chain with coefficients in a certain metric abelian group. NOTE: After this paper was posted, I learned that (1) and (2) were already proved by Choe and Morgan, respectively. Thus this paper is an exposition of their results.