  We propose a natural definition of the weighted $\sigma_k$-curvature for a
manifold with density; i.e.\ a triple $(M^n,g,e^{-\phi}\mathrm{dvol})$. This
definition is intended to capture the key properties of the
$\sigma_k$-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
$\sigma_k$-curvature functionals when the weighted $\sigma_k$-curvature is
variational. Second, we characterize the shrinking Gaussians as measures on
Euclidean space in terms of the total weighted $\sigma_k$-curvature
functionals. Third, we characterize when the weighted $\sigma_k$-curvature is
variational. These results are all analogues of their conformal counterparts,
and in the case $k=1$ recover some of the well-known properties of Perelman's
$\mathcal{W}$-functional.
