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The superposition principle for local 1-dimensional currents

Luigi Ambrosio, Federico Renzi, Federico Vitillaro

TL;DR

The paper extends Smirnov-type superposition principles to one-dimensional locally normal metric currents in complete metric spaces. By combining a conformal metric distortion with an isometric embedding into $\ell^\infty$, it reduces the problem to Paolini–Stepanov's decomposition for finite-mass currents, then reconstructs an integral representation $T=\int_{\Gamma(E)} T_\gamma \, d\eta(\gamma)$ with $\|T\|=\int_{\Gamma(E)} \|T_\gamma\| \, d\eta(\gamma)$ and a decomposition $T=A+C$ into acyclic and cyclic parts, while ensuring boundary mass compatibility. The approach introduces an auxiliary topology on open-ended curves $\Gamma(E)$, enabling a transport interpretation that can account for mass exchange with infinity, and proves injectivity of the acyclic part almost everywhere along the representing curves. The result broadens the scope of the superposition principle to local currents in Polish or complete metric spaces and provides a robust framework for transport and ODE-type analyses in non-smooth settings.

Abstract

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.

The superposition principle for local 1-dimensional currents

TL;DR

The paper extends Smirnov-type superposition principles to one-dimensional locally normal metric currents in complete metric spaces. By combining a conformal metric distortion with an isometric embedding into , it reduces the problem to Paolini–Stepanov's decomposition for finite-mass currents, then reconstructs an integral representation with and a decomposition into acyclic and cyclic parts, while ensuring boundary mass compatibility. The approach introduces an auxiliary topology on open-ended curves , enabling a transport interpretation that can account for mass exchange with infinity, and proves injectivity of the acyclic part almost everywhere along the representing curves. The result broadens the scope of the superposition principle to local currents in Polish or complete metric spaces and provides a robust framework for transport and ODE-type analyses in non-smooth settings.

Abstract

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.

Paper Structure

This paper contains 12 sections, 23 theorems, 130 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. (Local) metric currents
  4. Curves in metric spaces
  5. Cyclic and Acyclic currents
  6. Proof of the main result
  7. Plan of the proof
  8. Reduction to the case $\mathbf{E=\ell^\infty}$
  9. Conformal transformation and properties
  10. Extension of $\mathbf{T}$ to $\mathbf{(\overline{E}, \delta)}$
  11. An auxiliary topology on open ended curves with locally finite length in $E$
  12. Conclusion

Key Result

Theorem 1.4

Let $(E,d)$ be a complete metric space and let $T \in \mathbf{N}_{1,b}(E)$. Then there exists a positive Borel measure $\eta$ over $\Gamma(E)$ such that Moreover, $T$ can be decomposed as $T=A+C$, where $C$ is a cycle of $T$ and $A$ is acyclic. Finally, denoting by $\eta_A$ the measure associated to $A$ as in decomp and masseq, we have that $\eta_A$-almost every curve is injective, and that

Figures (4)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:

Theorems & Definitions (75)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 2.1
  • Definition 2.2: $0$-dimensional metric currents
  • Definition 2.3: Metric currents with finite and locally finite mass
  • Remark 2.4: Extensions of currents
  • Definition 2.5: Mass and support of currents
  • ...and 65 more