Curvature-Dimension Condition Meets Gromov's $n$-Volumic Scalar Curvature

Abstract

We study the properties of the $n$-volumic scalar curvature in this note. Lott-Sturm-Villani's curvature-dimension condition $\mathrm{CD}(\kappa ,n)$ was showed to imply Gromov's $n$-volumic scalar curvature $\ge n\kappa$ under an additional $n$-dimensional condition and we show the stability of $n$-volumic scalar curvature $\ge \kappa$ with respect to smGH-convergence. Then we propose a new weighted scalar curvature on the weighted Riemannian manifold and show its properties.