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Characterising 1-rectifiable metric spaces via connected tangent spaces

David Bate, Phoebe Valentine

TL;DR

The paper proves that a $1$-dimensional set $E$ in a complete metric space with finite $\mathcal{H}^1(E)$ and positive lower density is $1$-rectifiable if and only if almost every tangent space at a.e. point is connected, or equivalently is contained in the class of tangents with connected support $\mathcal{C}^*$ (or reduces to a single line tangent). The authors develop a novel Besicovitch-partition framework to extend circle-pair arguments from Euclidean settings to general metric spaces and show that connected tangents imply quasi-path connectivity of tangents. They combine tangent-measure convergence, density arguments, and the Besicovitch partitions to rule out purely unrectifiable portions, thereby obtaining the rectifiability conclusion. Overall, the work strengthens the tangent-based criteria for 1-rectifiability by replacing the need for bi-Lipschitz tangent structure with connected tangents, broadening applicability to general metric measure spaces and offering a sharp, density-aware characterization.

Abstract

We prove that in a complete metric space $X$, $1$-rectifiability of a set $E\subset X$ with $\mathcal{H}^1(E)<\infty$ and positive lower density $\mathcal{H}^1$-a.e. is implied by the property that all tangent spaces are connected metric spaces.

Characterising 1-rectifiable metric spaces via connected tangent spaces

TL;DR

The paper proves that a -dimensional set in a complete metric space with finite and positive lower density is -rectifiable if and only if almost every tangent space at a.e. point is connected, or equivalently is contained in the class of tangents with connected support (or reduces to a single line tangent). The authors develop a novel Besicovitch-partition framework to extend circle-pair arguments from Euclidean settings to general metric spaces and show that connected tangents imply quasi-path connectivity of tangents. They combine tangent-measure convergence, density arguments, and the Besicovitch partitions to rule out purely unrectifiable portions, thereby obtaining the rectifiability conclusion. Overall, the work strengthens the tangent-based criteria for 1-rectifiability by replacing the need for bi-Lipschitz tangent structure with connected tangents, broadening applicability to general metric measure spaces and offering a sharp, density-aware characterization.

Abstract

We prove that in a complete metric space , -rectifiability of a set with and positive lower density -a.e. is implied by the property that all tangent spaces are connected metric spaces.

Paper Structure

This paper contains 11 sections, 32 theorems, 138 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Measure Theory
  4. Rectifiability
  5. Density and Doubling
  6. Pointed Gromov-Hausdorff distance
  7. Convergence of measures
  8. Tangents of metric measure spaces
  9. Connected Tangents of Metric Spaces
  10. Besicovitch Partitions
  11. Connected Tangents Implies Rectifiable

Key Result

Theorem 1.1

Let $(X,d)$ be a complete metric space and let $E\subset X$ be $\mathcal{H}^1$-measurable with $\mathcal{H}^1(E)<\infty$. The following are equivalent:

Figures (2)

  • Figure 1: The contradiction in the proof of Lemma \ref{['lemma:uniformquasi']}. On the left, the limit of the sequence $(\nu_n,b_n)$ must have a radius which contains a $\delta$-quasi-path. On the right, for sufficiently large $n$, the existence of the $\delta$-quasi-path implies there must be points of $\operatorname{supp}\nu_n$ in the gap.
  • Figure 2: Step 2 of the proof of Proposition \ref{['lemma:besicovtichpartitions']}: Finding the ball in $\mathcal{B}_k$ that corresponds to $\text{Base}\,{\mathcal{A}_k}$.

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 2.1: Hausdorff measure
  • Lemma 2.2
  • proof
  • Definition 2.3: Lipschitz Map
  • Definition 2.4: Rectifiable
  • Lemma 2.5
  • proof
  • Definition 2.6: Density
  • Theorem 2.7: book:mattila Theorem 6.2
  • ...and 57 more