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Invariant Measures and Lower Ricci Curvature Bounds

  • Jaime Santos-RodríguezEmail author
Article
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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 GIso(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.

Keywords

Ricci curvature dimension Metric measure space Isometry group 

Mathematics Subject Classification (2010)

53C23 53C21 

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Notes

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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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversidad Autónoma de Madrid and ICMAT CSIC-UAM-UCM-UC3MMadridSpain

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