A notion of the weighted σk-curvature for manifolds with density

We propose a natural definition of the weighted σkσk-curvature for a manifold with density; i.e. a triple (Mn,g,e−ϕdvol)(Mn,g,e−ϕdvol). This definition is intended to capture the key properties of the σkσk-curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointwise changes of the measure. We justify our definition through three main results. First, we show that shrinking gradient Ricci solitons are local extrema of the total weighted σkσk-curvature functionals when the weighted σkσk-curvature is variational. Second, we characterize the shrinking Gaussians as measures on Euclidean space in terms of the total weighted σkσk-curvature functionals. Third, we characterize when the weighted σkσk-curvature is variational. These results are all analogues of their conformal counterparts, and in the case k=1k=1 recover some of the well-known properties of Perelman's WW-functional.