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 CD(κ, 𝑛) was shown to imply Gromov's 𝑛-volumic scalar curvature ≥ 𝑛κ under an additional 𝑛-dimensional condition, and we show the stability of 𝑛-volumic scalar curvature ≥ κ with respect to smGH-convergence. Then we propose a new weighted scalar curvature on the weighted Riemannian manifold and show its properties.