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2016

Behaviour of the reference measure on $\sf RCD$ spaces under charts

Gigli, Nicola, Pasqualetto, Enrico

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Mondino and Naber recently proved that finite dimensional $\sf RCD$ 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 $\sf RCD$ spaces.
A key tool that we use is a recent paper by De Philippis-Rindler about the structure of measures on the Euclidean space.

Behaviour of the reference measure on $\sf RCD$ spaces under charts · Around

Built on

J. Cheeger
1999
Earlier work this paper cites.
J. Cheeger and T. H. Colding
2000
Earlier work this paper cites.
N. Weaver
2000
Earlier work this paper cites.
K.-T. Sturm
2006
Earlier work this paper cites.

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2012
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2013
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Accepted at Mem. Amer. Math. Soc., arXiv:1407.0809, 2014
Original
N. Gigli · 2014
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Preprint, arXiv:1607.02554
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G. De Philippis, A. Marchese, and F. Rindler
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Accepted at Annals of Math. arXiv: 1601.06543
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Cited in the paper.
Preprint, arXiv:1607.02036
Original
M. Kell and A. Mondino
Cited in the paper.

Then

Preprint, arXiv:1405.2222, 2014
Original
A. Mondino and A. Naber · 2014
Later among the works it cites.
arXiv:1205.6622
Original
2015
Later among the works it cites.
G. Alberti and A. Marchese
2016
Closest in time.
N. Gigli, T. Rajala, and K.-T. Sturm
2016
Closest in time.

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