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...
Abstract: Since the scalar curvature appears in the term of Einstein field equations, the study of it becomes also important in general relativity. One of the open question about it is how to define the non-negative scalar curvature on non-smooth spaces. We will propose two definitions: one is the MV-scalar curvature on a closed topological manifold and the other one is n-volumic scalar curvatu...
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