Lower $N$-weighted Ricci curvature bound with $\varepsilon$-range and displacement convexity of entropies

TL;DR

This paper characterizes lower N-weighted Ricci curvature bounds with epsilon-range via entropy convexity in Wasserstein space, and derives related interpolation and functional inequalities.

Contribution

It introduces a new characterization of Ricci curvature bounds using entropy convexity with epsilon-range, extending previous frameworks.

Findings

Characterization of Ricci curvature bounds via entropy convexity.

Derivation of interpolation inequalities.

Development of functional inequalities.

Abstract

In the present article, we provide a characterization of a lower $N$-weighted Ricci curvature bound for $N \in ]-\infty,1]\cup[n,+\infty]$ with $\varepsilon$-range introduced by Lu-Minguzzi-Ohta in terms of a convexity of entropies over Wasserstein space. We further derive various interpolation inequalities and functional inequalities.