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

Milestones

Received: 12 January 2017

Revised: 3 June 2018

Accepted: 7 October 2018

Published: 21 May 2019