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Characterizations of the Upper Bound of Bakry–Emery Curvature

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Abstract

In this paper, we will present characterizations for the upper bound of the Bakry–Emery curvature on a Riemannian manifold by using functional inequalities on the path space. Moreover, characterizations for general lower and upper bounds of Ricci curvature are also given, which extends the recent results derived by Naber (Characterizations of bounded Ricci curvature on smooth and nonsmooth spaces, arXiv:1306.6512v4) and Wang–Wu (Sci China Math 61:1407–1420, 2018). A crucial point of the present study is to use a symmetrization argument for the lower and upper bounds of Ricci curvature, and a localization technique.

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Acknowledgements

It is a pleasure to thank M. Ledoux, F.Y. Wang, and X. Chen for useful conversations and the referee for useful suggestions. The author also thanks L.J. Cheng and A. Thalmaier who presented the draft of their paper [8]. This research is supported by NNSFC (11371099).

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Bo Wu

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

    Bo Wu

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Correspondence to Bo Wu.

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Wu, B. Characterizations of the Upper Bound of Bakry–Emery Curvature. J Geom Anal 30, 3923–3947 (2020). https://doi.org/10.1007/s12220-019-00222-2

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