[Paper Review] $(K,N)$-convexity and the curvature-dimension condition for negative $N$

Shin‐ichi Ohta|arXiv (Cornell University)|Oct 30, 2013

Geometric Analysis and Curvature Flows32 references4 citations

TL;DR

This paper extends the $(K,N)$-convexity framework, weighted Ricci curvature $\mathrm{Ric}_N$, and curvature-dimension condition $\mathrm{CD}(K,N)$ to negative values of $N$, generalizing key results such as Bochner's inequality, the Brunn–Minkowski inequality, and the equivalence between $\mathrm{Ric}_N \geq K$ and $\mathrm{CD}(K,N)$ to the negative $N$ regime. It establishes new functional inequalities, including a nontrivial $N$-Talagrand and $N$-log-Sobolev inequality for $K>0$, $N<0$, under the entropic curvature-dimension condition $\mathrm{CD}^e(K,N)$.

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 $Ric_N$ and the curvature-dimension condition $CD(K,N)$. We generalize a number of results in the case of $N&gt;0$ to this setting, including Bochner's inequality, the Brunn--Minkowski inequality and the equivalence between $Ric_N \ge K$ and $CD(K,N)$. We also show an expansion bound for gradient flows of Lipschitz $(K,N)$-convex functions.

Motivation & Objective

• To extend the theory of $(K,N)$-convexity, weighted Ricci curvature $\mathrm{Ric}_N$, and curvature-dimension condition $\mathrm{CD}(K,N)$ to negative values of $N$, which were previously restricted to $N>0$.
• To generalize fundamental results such as Bochner's inequality, the Brunn–Minkowski inequality, and the equivalence between $\mathrm{Ric}_N \geq K$ and $\mathrm{CD}(K,N)$ to the case $N<0$.
• To establish new functional inequalities—specifically, an $N$-Talagrand and an $N$-log-Sobolev inequality—under the entropic curvature-dimension condition $\mathrm{CD}^e(K,N)$ for $K>0$, $N<0$.
• To provide a rigorous framework for gradient flows of Lipschitz $(K,N)$-convex functions in the negative $N$ setting, including an expansion bound for such flows.

Proposed method

• Introduces the definition of $(K,N)$-convex functions on Riemannian manifolds and metric spaces, extending the framework of Erbar–Kuwada–Sturm to $N<0$.
• Derives an evolution variational inequality along gradient curves in the Riemannian setting, which serves as a foundation for regularizing estimates.
• Establishes an expansion bound for gradient flows of Lipschitz $(K,N)$-convex functions on Riemannian manifolds via the evolution variational inequality (Theorem 3.8).
• Generalizes Bochner's inequality and the Lichnerowicz inequality to negative $N$ using the weighted Ricci curvature $\mathrm{Ric}_N$.
• Defines the curvature-dimension condition $\mathrm{CD}(K,N)$ for $N<0$ and proves its equivalence to $\mathrm{Ric}_N \geq K$ in the Riemannian setting (Theorem 4.10).
• Applies the entropic curvature-dimension condition $\mathrm{CD}^e(K,N)$ to derive functional inequalities, including the $N$-Talagrand and $N$-log-Sobolev inequalities (Corollaries 4.18 and 4.19), using the geodesic convexity of the $E_N$-functional.

Experimental results

Research questions

• RQ1 Can the $(K,N)$-convexity framework be meaningfully extended to negative values of $N$?
• RQ2 Does the equivalence between $\mathrm{CD}(K,N)$ and $\mathrm{Ric}_N \geq K$ hold for $N<0$?
• RQ3 Can functional inequalities such as the Talagrand and log-Sobolev inequalities be generalized to the negative $N$ regime?
• RQ4 What is the behavior of gradient flows of $(K,N)$-convex functions when $N<0$?
• RQ5 How do the geodesic convexity properties of the $E_N$-functional lead to new inequalities in the negative $N$ case?

Key findings

• The paper proves that Bochner's inequality holds for $N<0$, extending a key analytic tool to the negative dimension regime.
• The equivalence between $\mathrm{CD}(K,N)$ and $\mathrm{Ric}_N \geq K$ is established for $N<0$, generalizing a central result from the positive $N$ case.
• An expansion bound for gradient flows of Lipschitz $(K,N)$-convex functions is derived, providing regularizing estimates in the negative $N$ setting (Theorem 3.8).
• A nontrivial $N$-Talagrand inequality is proven: $\mathrm{Ent}_{\mathfrak{m}}(\mu) \geq -N \log \left[ \cosh\left( \sqrt{-K/N} \, W_2(\mathfrak{m},\mu) \right) \right] $ for $K>0$, $N<0$, under $\mathrm{CD}^e(K,N)$.
• An $N$-log-Sobolev inequality is established: $KN \left( \exp(2\mathrm{Ent}_{\mathfrak{m}}(\mu)/N) - 1 \right) \leq I_{\mathfrak{m}}(\mu)$, valid under a positivity condition on the geodesic $E_N$-functional (Corollary 4.19).
• The paper shows that the $E_N$-functional satisfies a convexity inequality along minimal geodesics, leading to the derivation of the above functional inequalities via asymptotic analysis of the derivative at $t=0$.