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On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 1

Katrin Fässler, Ivan Yuri Violo

Abstract

We introduce new flatness coefficients, which we call $ι$-numbers, for Ahlfors $k$-regular sets in metric spaces ($k\in \mathbb{N}$). Using these coefficients for $k=1$, we characterize uniform $1$-rectifiability in rather general metric spaces, completing earlier work by Hahlomaa and Schul. Our proof proceeds by quantifying an isometric embedding theorem due to Menger, and by an abstract argument that allows to pass from a local covering by continua to a global covering by $1$-regular connected sets.

On various Carleson-type geometric lemmas and uniform rectifiability in metric spaces: Part 1

Abstract

We introduce new flatness coefficients, which we call -numbers, for Ahlfors -regular sets in metric spaces (). Using these coefficients for , we characterize uniform -rectifiability in rather general metric spaces, completing earlier work by Hahlomaa and Schul. Our proof proceeds by quantifying an isometric embedding theorem due to Menger, and by an abstract argument that allows to pass from a local covering by continua to a global covering by -regular connected sets.
Paper Structure (22 sections, 29 theorems, 205 equations)

This paper contains 22 sections, 29 theorems, 205 equations.

Table of Contents

  1. Introduction
  2. From $\beta$-numbers in Euclidean spaces to $\iota$-numbers in metric spaces
  3. From Menger curvature to $\iota$-numbers in dimension $1$
  4. Relation to previous work
  5. Euclidean-type (quantitative) rectifiability.
  6. Other notions of quantitative rectifiability
  7. Axiomatic results in metric spaces
  8. Approximate isometries
  9. Preliminaries
  10. Standard quantitative notions
  11. Ahlfors regular sets and dyadic systems
  12. Geometric lemmas for various coefficient functions
  13. New coefficients to measure flatness in metric spaces
  14. Sufficient conditions for the existence of regular covering curves
  15. Construction of $1$-regular covering continua
  16. ...and 7 more sections

Key Result

Theorem 1.4

Let $({\rm X},{\sf d})$ be a complete, doubling, and quasiconvex metric space. The following conditions are quantitatively equivalent for a $1$-regular set $E$ in $({\rm X},{\sf d})$:

Theorems & Definitions (63)

  • Theorem 1.4: Characterizations of uniform $1$-rectifiability
  • Definition 2.1: Quasiconvex metric space
  • Definition 2.2: Doubling metric space
  • Definition 2.3: $s$-regular sets
  • Definition 2.14: Geometric lemma
  • Remark 2.16
  • Example 2.17: $\beta$-numbers
  • Example 2.21: $\kappa$-numbers
  • Lemma 2.23: Different neighborhoods of cubes
  • proof
  • ...and 53 more