Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915440225157120 |
|---|---|
| author | Arroyo-Rabasa, Adolfo Bouchitté, Guy |
| author_facet | Arroyo-Rabasa, Adolfo Bouchitté, Guy |
| contents | A comprehensive study of one-dimensional metric currents and their relationship to the geometry of metric spaces is presented. We resolve the one-dimensional flat chain conjecture in this general setting, by proving that its validity is equivalent to a simple geometric connectedness property. More precisely, we prove that metric currents can be approximated in the mass norm by normal currents if and only if every $1$-rectifiable set can be covered by countably many Lipschitz curves up to an $\mathscr{H}^1$-negligible set. Building on this, we demonstrate that any $1$-current in a Banach space can be completed into a cycle by a rectifiable current, with the added mass controlled by the Kantorovich--Rubinstein norm of its boundary. We further refine our approximation result by showing that these currents can be approximated by polyhedral currents modulo a cycle. Finally, in arbitrary complete metric spaces, we establish a Smirnov-type decomposition for one-dimensional currents. This decomposition expresses such currents as a superposition, without mass cancellation, of currents associated with curves of bounded variation that have a vanishing Cantor part. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08212 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture Arroyo-Rabasa, Adolfo Bouchitté, Guy Analysis of PDEs Functional Analysis Metric Geometry 32C30, 49Q15, 51F30 (primary), 28A75 (secondary) A comprehensive study of one-dimensional metric currents and their relationship to the geometry of metric spaces is presented. We resolve the one-dimensional flat chain conjecture in this general setting, by proving that its validity is equivalent to a simple geometric connectedness property. More precisely, we prove that metric currents can be approximated in the mass norm by normal currents if and only if every $1$-rectifiable set can be covered by countably many Lipschitz curves up to an $\mathscr{H}^1$-negligible set. Building on this, we demonstrate that any $1$-current in a Banach space can be completed into a cycle by a rectifiable current, with the added mass controlled by the Kantorovich--Rubinstein norm of its boundary. We further refine our approximation result by showing that these currents can be approximated by polyhedral currents modulo a cycle. Finally, in arbitrary complete metric spaces, we establish a Smirnov-type decomposition for one-dimensional currents. This decomposition expresses such currents as a superposition, without mass cancellation, of currents associated with curves of bounded variation that have a vanishing Cantor part. |
| title | Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture |
| topic | Analysis of PDEs Functional Analysis Metric Geometry 32C30, 49Q15, 51F30 (primary), 28A75 (secondary) |
| url | https://arxiv.org/abs/2508.08212 |