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The superposition principle for local 1-dimensional currents

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arxiv 2503.18157 v1 pith:ZLABNLJ7 submitted 2025-03-23 math.MG math.APmath.FA

Luigi Ambrosio , Federico Renzi , Federico Vitillaro This is my paper · ORCID
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keywords currentslocallymetricnormalresultspacesadmitsambrosio-kirchheim
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We prove that every one-dimensional locally normal metric current, intended in the sense of U. Lang and S. Wenger, admits a nice integral representation through currents associated to (possibly unbounded) curves with locally finite length, generalizing the result shown by E. Paolini and E. Stepanov in the special case of Ambrosio-Kirchheim normal currents. Our result holds in Polish spaces, or more generally in complete metric spaces for 1-currents with tight support.

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  1. Structure of Metric $1$-currents: approximation by normal currents and representation results

    math.MG 2025-08 conditional novelty 8.0 of 10

    Every metric 1-current in a complete quasiconvex metric space is a mass limit of normal 1-currents, and in complete separable spaces it is an integral of curve fragments.

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