The weighted ${\sigma }_k$-curvature of a smooth metric measure space

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.

52 pages

References in corpus (1)

• A notion of the weighted ${\sigma }_k$-curvature for manifolds with density

Cited by in corpus (2)

• Local pointwise second derivative estimates for strong solutions to the ${\sigma }_k$-Yamabe equation on Euclidean domains
• The weighted ambient metric for manifolds with density