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Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE

David Bate, Julian Weigt

TL;DR

The paper establishes that in a complete metric space, a Borel set $E$ with $ ext{H}^n(E)< $ is $n$-rectifiable provided $ ext{H}^n|_E$ supports $n$ independent Alberti representations, strengthening prior density-based results. It develops a quantitative higher-integrability theory for measures constrained by PDEs, extending De Philippis–Rindler to general operators via a Calderón–Zygmund framework and PDE decompositions. By encoding measures as disintegrations along curve fragments and refining Alberti representations, the authors derive lower-density estimates at multiple scales and connect these to rectifiability through a doubling-dichotomy and scale induction. The results have broad implications for rectifiability in metric spaces, including consequences for Lipschitz differentiability spaces and metric currents, and provide quantitative tools for analyzing PDE-constrained measures in geometric measure theory. Overall, the work advances the understanding of when geometric regularity (rectifiability) follows from curve decomposability in metric spaces, with independent interest in the higher integrability of PDE-governed measures.

Abstract

We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with $\mathcal{H}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an Alberti representation. Precisely, we show that if $\mathcal{H}^n|_E$ has $n$ independent Alberti representations, then $E$ is $n$-rectifiable. This is a sharp strengthening of prior results of Bate and Li. It has been known for some time that such a result answers many open questions concerning rectifiability in metric spaces, which we discuss. An important step of our proof is to establish the higher integrability of measures on Euclidean space satisfying a PDE constraint. These results provide a quantitative generalisation of recent work of De Philippis and Rindler and are of independent interest.

Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE

TL;DR

The paper establishes that in a complete metric space, a Borel set with is -rectifiable provided supports independent Alberti representations, strengthening prior density-based results. It develops a quantitative higher-integrability theory for measures constrained by PDEs, extending De Philippis–Rindler to general operators via a Calderón–Zygmund framework and PDE decompositions. By encoding measures as disintegrations along curve fragments and refining Alberti representations, the authors derive lower-density estimates at multiple scales and connect these to rectifiability through a doubling-dichotomy and scale induction. The results have broad implications for rectifiability in metric spaces, including consequences for Lipschitz differentiability spaces and metric currents, and provide quantitative tools for analyzing PDE-constrained measures in geometric measure theory. Overall, the work advances the understanding of when geometric regularity (rectifiability) follows from curve decomposability in metric spaces, with independent interest in the higher integrability of PDE-governed measures.

Abstract

We give a sufficient condition for a Borel subset of a complete metric space with to be -rectifiable. This condition involves a decomposition of into rectifiable curves known as an Alberti representation. Precisely, we show that if has independent Alberti representations, then is -rectifiable. This is a sharp strengthening of prior results of Bate and Li. It has been known for some time that such a result answers many open questions concerning rectifiability in metric spaces, which we discuss. An important step of our proof is to establish the higher integrability of measures on Euclidean space satisfying a PDE constraint. These results provide a quantitative generalisation of recent work of De Philippis and Rindler and are of independent interest.
Paper Structure (17 sections, 33 theorems, 172 equations)

This paper contains 17 sections, 33 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Rectifiability of Lipschitz differentiability spaces and metric currents
  3. Weak tangent fields and Lipschitz projections
  4. Outline of the proof of the main theorem
  5. Acknowledgements
  6. Higher integrability of measures satisfying a PDE
  7. The divergence case
  8. General differential operator
  9. Measures defined by disintegration and Alberti representations
  10. Hausdorff distance and the measurability of the intersection operator
  11. Curve fragments
  12. Defining measures via disintegration
  13. Application of Theorem 1.9
  14. Alberti representations
  15. Approximating fragments by curves
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $X$ be a complete metric space and let $E\subset X$ be Borel with $\mathcal{H}^n(E)<\infty$ such that $\mathcal{H}^n{\mathbin\upharpoonright}_{E}$ has $n$ independent Alberti representations. Then $E$ is $n$-rectifiable.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 74 more