Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces

TL;DR

This paper investigates the geometric structure of Ahlfors regular Lipschitz differentiability spaces, showing that maximal dimension charts lead to uniformly rectifiable tangents, while non-maximal charts are strongly unrectifiable.

Contribution

It establishes a dichotomy in the rectifiability of tangents based on the dimension of charts in Lipschitz differentiability spaces.

Findings

Maximal dimension charts imply uniformly rectifiable tangents.

Almost every point admits a tangent isometric to a finite-dimensional Banach space.

Non-maximal dimension charts are strongly unrectifiable.

Abstract

We study Lipschitz differentiability spaces, a class of metric measure spaces introduced by Cheeger. We show that if an Ahlfors regular Lipschitz differentiability space has charts of maximal dimension, then, at almost every point, all its tangents are uniformly rectifiable. In particular, at almost every point, such a space admits a tangent that is isometric to a finite-dimensional Banach space. In contrast, we also show that if an Ahlfors regular Lipschitz differentiability space has charts of non-maximal dimension, then these charts are strongly unrectifiable in the sense of Ambrosio-Kirchheim.