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Curvature bounds for configuration spaces

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Abstract

We show that the configuration space \(\Upsilon \) over a manifold \(M\) inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on \(M\) implies a lower Ricci curvature bound on \(\Upsilon \) in the sense of Lott–Sturm–Villani, the Bochner inequality, gradient estimates and Wasserstein contraction. Moreover, we show that the heat flow on \(\Upsilon \) can be identified as the gradient flow of the entropy.

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Notes

  1. The results also hold in the case that \(M\) is compact. However, they can be derived much easier.

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Acknowledgments

The authors would like to thank Theo Sturm and Fabio Cavalletti for several fruitful discussions on the subject of this paper.

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Authors and Affiliations

  1. Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Matthias Erbar & Martin Huesmann

Authors
  1. Matthias Erbar
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  2. Martin Huesmann
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Corresponding author

Correspondence to Matthias Erbar.

Additional information

Communicated by L. Ambrosio.

This material is based upon work supported by the National Science Foundation under Grant No. 0932078 000, while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, in the fall of 2013. M.E. gratefully acknowledges funding from the European Research Council under the European Communitys Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement GeMeThnES No. 246923; M.H. from the CRC 1060.

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Erbar, M., Huesmann, M. Curvature bounds for configuration spaces. Calc. Var. 54, 397–430 (2015). https://doi.org/10.1007/s00526-014-0790-1

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