Flat norm decomposition of integral currents

Abstract

Currents represent generalized surfaces studied in geometric measure theory. They range from relatively tame integral currents representing oriented compact manifolds with boundary and integer multiplicities, to arbitrary elements of the dual space of differential forms. The flat norm provides a natural distance in the space of currents, and works by decomposing a $d$-dimensional current into $d$- and (the boundary of) $(d+1)$-dimensional pieces in an optimal way. Given an integral current, can we expect its flat norm decomposition to be integral as well? This is not known in general, except in the case of $d$-currents that are boundaries of $(d+1)$-currents in $\mathbb{R}^{d+1}$ (following results from a corresponding problem on the $L^1$ total variation ($L^1$TV) of functionals). On the other hand, for a discretized flat norm on a finite simplicial complex, the analogous statement holds even when the inputs are not boundaries. This simplicial version relies on the total unimodularity of the boundary matrix of the simplicial complex -- a result distinct from the $L^1$TV approach. We develop an analysis framework that extends the result in the simplicial setting to one for $d$-currents in $\mathbb{R}^{d+1}$, provided a suitable triangulation result holds. In $\mathbb{R}^2$, we use a triangulation result of Shewchuk (bounding both the size and location of small angles), and apply the framework to show that the discrete result implies the continuous result for $1$-currents in $\mathbb{R}^2$.