AbstractInspired by the importance of the Bakry-Emery curvature on a weighted Riemannian manifold $(M^n, g, e^{f}Vol_g)$, we will introduce the weighted scalar curvature on it and then extend some classic results of the scalar curvature to the weighted version. For example, we will generalize Schoen-Yau's minimal hypersurface method, Gromov-Lawson's index theory approach and Seiberg-Witten inva...
AbstractThe scalar curvature of a Riemannian metric is interesting not only in analysis, geometry, and topology, but also in physics. Enlargeable Length-structures will be introduced and showed that it is a new obstruction to the existence of a Riemannian metric with positive scalar curvature (PSC-metric). Thus, the connected sum of a closed manifold with some of locally CAT(0)-manifolds carry ...
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