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

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.