Bakry-\'Emery conditions on almost smooth metric measure spaces

TL;DR

This paper establishes a sufficient condition for almost smooth metric measure spaces to satisfy the Bakry-Émery condition, demonstrating that it is weaker than the RCD condition and providing examples with non-constant local dimension.

Contribution

It introduces a new sufficient condition for BE(K, N) on almost smooth spaces and shows that BE is weaker than RCD, with examples of spaces having non-constant local dimension.

Findings

Glued space of two Riemannian manifolds satisfies BE(K, N)

BE condition is weaker than RCD condition in this setting

First example of Ricci lower bound with non-constant local dimension

Abstract

In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-\'Emery condition $BE (K, N)$. The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the $BE$ condition is strictly weaker than the $RCD$ condition even in this setting, and that the local dimension is not constant even if the space satisfies the $BE$ condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-\'Emery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be $RCD(K, N)$ spaces.