Rectifiability of planes and Alberti representations

TL;DR

This paper investigates the structure of metric measure spaces with topological and differentiable properties, demonstrating that certain subsets are at most 2-rectifiable and have limited Alberti representations, advancing understanding of geometric measure theory.

Contribution

It establishes that subsets with blowups as topological planes can have at most two independent Alberti representations and characterizes their rectifiability, addressing a specific open question.

Findings

Subsets with blowups as topological planes admit at most 2 Alberti representations.

If such a subset admits 2 Alberti representations, it is 2-rectifiable.

Provides partial answers to a question on rectifiability in metric measure spaces.

Abstract

We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let $X$ be a pointwise doubling metric measure space. Let $U$ be a Borel subset on which the blowups of $X$ are topological planes. We show that $U$ can admit at most $2$ independent Alberti representations. Furthermore, if $U$ admits $2$ Alberti representations, then the restriction of the measure to $U$ is $2$-rectifiable. This is a partial answer to the case $n=2$ of a question of the second author and Schioppa.