Flat norm decompositions of integral currents

Currents represent generalized surfaces studied in geometric measure theory. They range from rel-atively tame integral currents that represent oriented manifolds with integer multiplicities and finite volume as well as boundary, 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. A natural question about currents is the following. If the input is an integral current, can its flat norm decomposition be integral as well? The answer is not known in general, except in the case of d-currents that are boundaries of (d+ 1)-currents in Rd+1. Following correspondences between the flat norm and L1 total variation (L1TV) of functionals, the answer is affirmative in this case. On the other hand, for the discretization of the flat norm on a finite simplicial complex, the analogous statement remains true even when the inputs are not boundaries. This result is implied by the boundary matrix of the simplicial complex being totally unimodular – a result distinct from the L1TV approach. We develop an analysis framework that extends the result in the simplicial setting to that for d-currents in Rd+1, provided a suitable triangulation result holds. Following results of Shewchuk on triangulating planar straight line graphs while bounding both the size and location of small angles, our framework shows that the discrete result implies the continuous result for the case of 1-currents in R2.