$(K,N)$-convexity and the curvature-dimension condition for negative $N$

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 $Ri{c}_N$ and the curvature-dimension condition $CD(K,N)$. 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 $Ri{c}_N\ge K$ and $CD(K,N)$. We also show an expansion bound for gradient flows of Lipschitz $(K,N)$-convex functions.

29 pages; Minor corrections, to appear in J. Geom. Anal

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Cited by in corpus (3)

• On Interpolation and Curvature via Wasserstein Geodesics
• Positive weighted sectional curvature
• Sectional curvature for Riemannian manifolds with density