Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An atomic decomposition of one-dimensional metric currents without boundary

You-Wei Benson Chen, Jesse Goodman, Felipe Hernandez, Daniel Spector

TL;DR

The paper develops a rigorous atomic decomposition for 1-currents without boundary in geodesic metric spaces, representing any divergence-free current as a limit of weighted sums of closed Lipschitz curves with uniformly controlled Morrey norms. The approach hinges on two pillars: (i) representing currents as integrals over closed piecewise-geodesic curves and (ii) a Surgery Lemma that decomposes a closed piecewise-geodesic curve into a finite collection of closed curves with near-length preservation and uniform Morrey bound. This yields a concrete, curve-based atomic structure with explicit mass and Morrey-scale controls, refining the Euclidean theory of divergence-free measures via polygonal approximations. The results bridge geometric measure theory on metric spaces with Hardy-type atomic frameworks, and they provide a pathway to studying PDEs on manifolds by enabling stable, Morrey-bounded curve decompositions of 1-currents. Overall, the work significantly advances the understanding of how 1-currents without boundary can be synthesized from one-dimensional, Morrey-controlled atoms, with implications for analysis on metric spaces and divergence-free measures in Euclidean settings.

Abstract

This paper proves an atomic decomposition of the space of $1$-dimensional metric currents without boundary, in which the atoms are specified by closed Lipschitz curves with uniform control on their Morrey norms. Our argument relies on a geometric construction which states that for any $ε>0$ one can express a piecewise-geodesic closed curve as the sum of piecewise-geodesic closed curves whose total length is at most $(1+ε)$ times the original length and whose Morrey norms are each bounded by a universal constant times $ε^{-2}$. In Euclidean space, our results refine the state of the art, providing an approximation of divergence free measures by limits of sums of closed polygonal paths whose total length can be made arbitrarily close to the norm of the approximated measure.

An atomic decomposition of one-dimensional metric currents without boundary

TL;DR

The paper develops a rigorous atomic decomposition for 1-currents without boundary in geodesic metric spaces, representing any divergence-free current as a limit of weighted sums of closed Lipschitz curves with uniformly controlled Morrey norms. The approach hinges on two pillars: (i) representing currents as integrals over closed piecewise-geodesic curves and (ii) a Surgery Lemma that decomposes a closed piecewise-geodesic curve into a finite collection of closed curves with near-length preservation and uniform Morrey bound. This yields a concrete, curve-based atomic structure with explicit mass and Morrey-scale controls, refining the Euclidean theory of divergence-free measures via polygonal approximations. The results bridge geometric measure theory on metric spaces with Hardy-type atomic frameworks, and they provide a pathway to studying PDEs on manifolds by enabling stable, Morrey-bounded curve decompositions of 1-currents. Overall, the work significantly advances the understanding of how 1-currents without boundary can be synthesized from one-dimensional, Morrey-controlled atoms, with implications for analysis on metric spaces and divergence-free measures in Euclidean settings.

Abstract

This paper proves an atomic decomposition of the space of -dimensional metric currents without boundary, in which the atoms are specified by closed Lipschitz curves with uniform control on their Morrey norms. Our argument relies on a geometric construction which states that for any one can express a piecewise-geodesic closed curve as the sum of piecewise-geodesic closed curves whose total length is at most times the original length and whose Morrey norms are each bounded by a universal constant times . In Euclidean space, our results refine the state of the art, providing an approximation of divergence free measures by limits of sums of closed polygonal paths whose total length can be made arbitrarily close to the norm of the approximated measure.

Paper Structure

This paper contains 21 sections, 19 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Discussion
  4. The atoms of the decomposition
  5. Interpreting the Morrey norm
  6. A comparison with classical atoms and dimension stability
  7. Overview of the proof
  8. Preliminaries
  9. Lipschitz curves
  10. Geodesic curves
  11. Piecewise geodesic curves
  12. Closed curves and the basic cut operation
  13. The Surgery Lemma
  14. Regularity conditions on piecewise-geodesic curves
  15. Type I cut operation
  16. ...and 6 more sections

Key Result

Theorem 1.1

There is a universal constant $C$ such that if $E$ is a complete, separable, geodesic metric space, $T \in \mathcal{M}_1(E)$ satisfies $\partial T=0$, and $0<\varepsilon<1$, then there exist piecewise-geodesic closed curves $\gamma_{i,n}$ and scalars $\lambda_{i,n} \geq 0$, $n \in \mathbb{N}$, $1\le

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Lemma 3.1: Surgery
  • Corollary 3.2
  • ...and 32 more