Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 013, 20 pages      arXiv:2001.04087
Contribution to the Special Issue on Scalar and Ricci Curvature in honor of Misha Gromov on his 75th Birthday

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

Jialong Deng
Mathematisches Institut, Georg-August-Universität, Göttingen, Germany

Received July 29, 2020, in final form January 23, 2021; Published online February 05, 2021

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

Key words: curvature-dimension condition; $n$-volumic scalar curvature; stability; weighted scalar curvature ${\rm Sc}_{\alpha, \beta}$.

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