Characterizations of rectifiable metric measure spaces

TL;DR

This paper characterizes n-rectifiable metric measure spaces by decomposing them into parts that satisfy specific geometric and measure-theoretic conditions, using an iterative grid construction and Alberti representations.

Contribution

It introduces a new characterization of n-rectifiable spaces via Lipschitz differentiability, Alberti representations, and David's condition, with a novel iterative grid method.

Findings

Spaces with positive finite n-density can be decomposed into Lipschitz differentiability spaces.

The iterative grid construction shows images of high-density curve sets contain large balls.

The approach confirms biLipschitz pieces exist under the given conditions.

Abstract

We characterize $n$-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz differentiability space; has $n$-independent Alberti representations; satisfies David's condition for an $n$-dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisfies David's condition. This allows us to apply previously known "biLipschitz pieces" results on the charts.