Structure of Metric $1$-currents: approximation by normal currents and representation results
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913984585662464 |
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| author | Bate, David Caputo, Emanuele Takáč, Jakub Valentine, Phoebe Wald, Pietro |
| author_facet | Bate, David Caputo, Emanuele Takáč, Jakub Valentine, Phoebe Wald, Pietro |
| contents | We prove the $1$-dimensional flat chain conjecture in any complete and quasiconvex metric space, namely that metric $1$-currents can be approximated in mass by normal $1$-currents. The proof relies on a new Banach space isomorphism theorem, relating metric $1$-currents and their boundaries to the Arens-Eells space. As a by-product, any metric $1$-current in a complete and separable metric space can be represented as the integral superposition of oriented $1$-rectifiable sets, thus dropping a finite dimensionality condition from previous results of Schioppa [Schioppa Adv. Math. 2016, Schioppa J. Funct. Anal. 2016]. The connection between the flat chain conjecture and the representation result is provided by a structure theorem for metric $1$-currents in Banach spaces, showing that any such current can be realised as the restriction to a Borel set of a boundaryless normal $1$-current. This generalizes, to any Banach space, the $1$-dimensional case of a recent result of Alberti-Marchese in Euclidean spaces [Alberti-Marchese 2023]. The argument of Alberti-Marchese requires the strict polyhedral approximation theorem of Federer for normal $1$-currents, which we obtain in Banach spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure of Metric $1$-currents: approximation by normal currents and representation results Bate, David Caputo, Emanuele Takáč, Jakub Valentine, Phoebe Wald, Pietro Metric Geometry Analysis of PDEs Functional Analysis 49Q15, 28A75, 30L99 We prove the $1$-dimensional flat chain conjecture in any complete and quasiconvex metric space, namely that metric $1$-currents can be approximated in mass by normal $1$-currents. The proof relies on a new Banach space isomorphism theorem, relating metric $1$-currents and their boundaries to the Arens-Eells space. As a by-product, any metric $1$-current in a complete and separable metric space can be represented as the integral superposition of oriented $1$-rectifiable sets, thus dropping a finite dimensionality condition from previous results of Schioppa [Schioppa Adv. Math. 2016, Schioppa J. Funct. Anal. 2016]. The connection between the flat chain conjecture and the representation result is provided by a structure theorem for metric $1$-currents in Banach spaces, showing that any such current can be realised as the restriction to a Borel set of a boundaryless normal $1$-current. This generalizes, to any Banach space, the $1$-dimensional case of a recent result of Alberti-Marchese in Euclidean spaces [Alberti-Marchese 2023]. The argument of Alberti-Marchese requires the strict polyhedral approximation theorem of Federer for normal $1$-currents, which we obtain in Banach spaces. |
| title | Structure of Metric $1$-currents: approximation by normal currents and representation results |
| topic | Metric Geometry Analysis of PDEs Functional Analysis 49Q15, 28A75, 30L99 |
| url | https://arxiv.org/abs/2508.08017 |