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arXiv · 1608.01663

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

Jeffrey S. Case
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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.

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Jeffrey S. Case. 2016-08-04. The weighted $\sigma_k$-curvature of a smooth metric measure space. https://doi.org/10.2140/pjm.2019.299.339

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