Alberti representations and rectifiability of metric spaces

Abstract

We give a sufficient condition for a Borel subset $E\subset X$ of a complete metric space with ${\mathrm{H}}^n(E)<\infty$ to be $n$-rectifiable. This condition involves a decomposition of $E$ into rectifiable curves known as an \emph{Alberti representation}. Precisely, we show that if ${\mathrm{H}}^n{|}_E$ has $n$ independent Alberti representations, then $E$ is $n$-rectifiable. This is a sharp strengthening of prior results of Bate and Li. It has been known for some time that such a result answers many open questions concerning rectifiability in metric spaces, which we discuss.

arXiv comment1. Parts of the introduction have been shortened or rewritten. 2. The level of abstraction in the proof has been reduced in some places and the notation simplified, with the proof ideas remaining essentially identical. 3. The sections have been restructured. 4. The title has been adjusted and most secondary content regarding the quantitative PDE result has been removed