Deformation of the Weighted Scalar Curvature

Abstract

Inspired by the work of Fischer-Marsden [Duke Math. J. 42 (1975), 519-547], we study in this paper the deformation of the weighted scalar curvature. By studying the kernel of the formal $L_{\varphi }^2$-adjoint for the linearization of the weighted scalar curvature, we prove several geometric results. In particular, we define a weighted vacuum static space, and study locally conformally flat weighted vacuum static spaces. We then prove some stability results of the weighted scalar curvature on flat spaces. Finally, we consider the prescribed weighted scalar curvature problem on closed smooth metric measure spaces.

References in corpus (5)

• A note on static spaces and related problems
• Smooth metric measure spaces, quasi-Einstein metrics, and tractors
• Some constructions of formally self-adjoint conformally covariant polydifferential operators
• Deformation of the scalar curvature and the mean curvature
• Weighted renormalized volume coefficients