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Invariant Measures and Lower Ricci Curvature Bounds

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Abstract

Given a metric measure space \((X,d,\mathfrak {m})\) that satisfies the Riemannian Curvature Dimension condition, RCD∗(K,N), and a compact subgroup of isometries G ≤ Iso(X) we prove that there exists a G −invariant measure, \(\mathfrak {m}_G\) equivalent to \(\mathfrak {m}\) such that \((X,d,\mathfrak {m}_G)\) is still a RCD∗(K,N) space. We also obtain applications to Lie group actions on RCD∗(K,N) spaces. We look at homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed subgroups of isometries.

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Acknowledgements

The author would like to thank his advisor Prof. Luis Guijarro for helpful comments on earlier versions of this manuscript.

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  1. Department of Mathematics, Universidad Autónoma de Madrid and ICMAT CSIC-UAM-UCM-UC3M, Madrid, Spain

    Jaime Santos-Rodríguez

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  1. Jaime Santos-Rodríguez
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Correspondence to Jaime Santos-Rodríguez.

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The author was supported by research grants MTM2014-57769-C3-3-P, and MTM2017-85934-C3-2-P (MINECO) and ICMAT Severo Ochoa Project SEV-2015-0554-17-1 (MINECO).

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Santos-Rodríguez, J. Invariant Measures and Lower Ricci Curvature Bounds. Potential Anal 53, 871–897 (2020). https://doi.org/10.1007/s11118-019-09790-y

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