The weighted σ_k-curvature of a smooth metric measure space

We propose a definition of the weighted σ_k-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted σ_k-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when k = 1, 2 or the smooth metric measure space is locally conformally flat in the weighted sense. Second, we show that, in the variational cases, quasi-Einstein metrics are stable with respect to the total weighted σ_k-curvature functional. We also discuss related conjectures for weighted Einstein manifolds.