Uniformly rectifiable metric spaces: Lipschitz images, Bi-Lateral Weak Geometric Lemma and Corona Decompositions

TL;DR

This paper establishes the equivalence of key geometric conditions characterizing uniform rectifiability in Ahlfors regular metric spaces, extending classical Euclidean results to a broader metric setting.

Contribution

It proves the equivalence of Big Pieces of Lipschitz Images, Bi-lateral Weak Geometric Lemma, and Corona Decomposition in any Ahlfors regular metric space, generalizing foundational Euclidean concepts.

Findings

Proved equivalence of geometric conditions in metric spaces

Extended Euclidean rectifiability concepts to general metric spaces

Studied Reifenberg parameterizations in this context

Abstract

In their 1991 and 1993 foundational monographs, David and Semmes characterized uniform rectifiability for subsets of Euclidean space in a multitude of geometric and analytic ways. The fundamental geometric conditions can be naturally stated in any metric space and it has long been a question of how these concepts are related in this general setting. In this paper we prove their equivalence. Namely, we show the equivalence of Big Pieces of Lipschitz Images, Bi-lateral Weak Geometric Lemma and Corona Decomposition in any Ahlfors regular metric space. Loosely speaking, this gives a quantitative equivalence between having Lipschitz charts and approximations by nicer spaces. En route, we also study Reifenberg parameterizations.