Abstract

Inspired 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 invariants (in four dimensions) to a weighted Riemannian manifold with positively weighted scalar curvature.