Characterizations of the upper bound of Bakry-Emery curvature

TL;DR

This paper characterizes the upper bound of Bakry-Emery curvature on Riemannian manifolds using functional inequalities on path space, extending existing results on Ricci curvature bounds.

Contribution

It provides new characterizations for both upper and lower bounds of Ricci curvature, generalizing recent findings by Naber and Wang-Wu.

Findings

Characterizations of the upper bound of Bakry-Emery curvature.

Extensions to general bounds of Ricci curvature.

Use of functional inequalities on path space.

Abstract

In this paper, we will present some characterizations for the upper bound of the Bakry-Emery curvature on a Riemannian manifold by using functional inequalities on path space. Moreover, some characterizations for general lower and upper bounds of Ricci curvature are also given, which extends the recent results derived by Naber \cite{N} and Wang-Wu\cite{WW}.