#### Vol. 299, No. 2, 2019

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The weighted $\sigma_k$-curvature of a smooth metric measure space

### Jeffrey S. Case

Vol. 299 (2019), No. 2, 339–399
DOI: 10.2140/pjm.2019.299.339
##### Abstract

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

##### Keywords
smooth metric measure space, $\sigma_k$-curvature, quasi-Einstein, weighted Einstein
##### Mathematical Subject Classification 2010
Primary: 53C21
Secondary: 35J60, 58E11