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

TL;DR

This paper introduces a new definition of weighted $\sigma_k$-curvature for smooth metric measure spaces, analyzes the associated PDEs, and explores stability and conjectures related to weighted Einstein manifolds.

Contribution

It defines weighted $\sigma_k$-curvature, studies the governing PDEs, and examines stability of quasi-Einstein metrics within this new framework.

Findings

Weighted $\sigma_k$-curvature prescription problem is governed by a fully nonlinear elliptic PDE.

In variational cases, quasi-Einstein metrics are stable under the total weighted $\sigma_k$-curvature functional.

Discussion of conjectures for weighted Einstein manifolds.

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.