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(K, N)-Convexity and the Curvature-Dimension Condition for Negative N

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Abstract

We extend the range of N to negative values in the (K, N)-convexity (in the sense of Erbar–Kuwada–Sturm), the weighted Ricci curvature \(\mathop {\mathrm {Ric}}\nolimits _N\) and the curvature-dimension condition \(\mathop {\mathrm {CD}}\nolimits (K,N)\). We generalize a number of results in the case of \(N>0\) to this setting, including Bochner’s inequality, the Brunn–Minkowski inequality and the equivalence between \(\mathop {\mathrm {Ric}}\nolimits _N \ge K\) and \(\mathop {\mathrm {CD}}\nolimits (K,N)\). We also show an expansion bound for gradient flows of Lipschitz (K, N)-convex functions.

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Acknowledgments

I am grateful to Kazumasa Kuwada for valuable suggestions and discussions, especially on the expansion bound in Subsection 3.2. I thank Asuka Takatsu for fruitful discussions, some of the results in Subsections 4.1, 4.2 originate from discussions during the joint work [33, 34]. My gratitude also goes to Frank Morgan for drawing my attention to [25] and [18], and to Emanuel Milman for his helpful comments on the background of [25] and [18]. Supported in part by the Grant-in-Aid for Young Scientists (B) 23740048.

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  1. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan

    Shin-ichi Ohta

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  1. Shin-ichi Ohta
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Ohta, Si. (K, N)-Convexity and the Curvature-Dimension Condition for Negative N . J Geom Anal 26, 2067–2096 (2016). https://doi.org/10.1007/s12220-015-9619-1

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  • DOI: https://doi.org/10.1007/s12220-015-9619-1

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