The weighted \(\sigma_k\)-curvature of a smooth metric measure space (Q2420115)

In this paper, the author proposes a new definition of the weighted $\sigma_k$-curvature of a smooth metric measure space. In the category of Einstein metrics, an effective tool is to study the total $\sigma_k$-curvature and its variational characterizations, which is particularly useful in the classification problems. As a generalization, in the context of metric measure spaces and quasi-Einstein metrics, the author defines the weighted $\sigma_k$-curvature and studies its variational properties. The main results of the paper are summarized as follows. Theorem 1.1 gives a variational characterization for the the weighted $\sigma_k$-curvature. Theorem 1.2 concerns the second variation of weighted $\sigma_k$-functional $\mathcal{F}_k$. As a geometric application, Theorem 1.3 proves that, under the assumption of conformal flatness, quasi-Einstein manifolds are the only critical points of $\mathcal{F}_k$ in the weighted elliptic $k$-cones. In Theorem 1.4, the author proves that, in the case of $k=1,2$, quasi-Einstein manifolds and weighted Einstein manifolds are the only critical points of the weighted functionals $\mathcal{F}_k$ and $\mathcal{Y}_k$ respectively.