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

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.

arXiv comment17 pages

References in corpus (1)

• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces

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• A sufficient condition to a regular set of positive measure on RCD spaces
• On the structure of RCD spaces with upper curvature bounds
• On the isometry group of $RC{D}^{\ast }(K,N)$-spaces
• Collapsed Ricci limit spaces as non-collapsed $RCD$ spaces
• Monotonicity formulas for harmonic functions in $\mathrm{RCD}(0,N)$ spaces
• Representation theorems for normed modules
• On the asymptotic behavior of the dimension of spaces of harmonic functions with polynomial growth