Communications in Analysis and Geometry

Volume 29 (2021)

Number 6

Behaviour of the reference measure on $\mathsf{RCD}$ spaces under charts

Pages: 1391 – 1414

DOI: https://dx.doi.org/10.4310/CAG.2021.v29.n6.a3

Authors

Nicola Gigli (Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy)

Enrico Pasqualetto (Scuola Internazionale Superiore di Studi Avanzati, Trieste, Italy)

Abstract

Mondino and Naber recently proved that finite dimensional $\mathsf{RCD}$ spaces are rectifiable.

Here we show that the push-forward of the reference measure under the charts built by them is absolutely continuous with respect to the Lebesgue measure. This result, read in conjunction with another recent work of us, has relevant implications on the structure of tangent spaces to $\mathsf{RCD}$ spaces.

A key tool that we use is a recent paper by De Philippis–Rindler about the structure of measures on the Euclidean space.

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Received 29 November 2016

Accepted 8 May 2017

Published 11 January 2022