We propose a definition of the weighted σk-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted σk-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 σk-curvature functional. We also discuss related conjectures for weighted Einstein manifolds.
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