On a conjecture of Cheeger

TL;DR

This paper proves Cheeger's conjecture on the structure of Lipschitz differentiability spaces by leveraging a recent structure theorem for normal 1-currents, showing the measure's push-forward is absolutely continuous w.r.t. Lebesgue measure.

Contribution

It introduces a novel proof of Cheeger's conjecture using a recent structure theorem for normal 1-currents, establishing measure regularity in Lipschitz differentiability spaces.

Findings

The push-forward measure under a chart is absolutely continuous with respect to Lebesgue measure.

The proof relies on a recent structure theorem for normal 1-currents.

The result confirms a key conjecture in the structure theory of Lipschitz differentiability spaces.

Abstract

This note details how a recent structure theorem for normal $1$-currents proved by the first and third author allows to prove a conjecture of Cheeger concerning the structure of Lipschitz differentiability spaces. More precisely, we show that the push-forward of the measure from a Lipschitz differentiability space under a chart is absolutely continuous with respect to Lebesgue measure.