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

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

smooth metric measure space, $\sigma_k$-curvature, quasi-Einstein, weighted Einstein
Mathematical Subject Classification 2010
Primary: 53C21
Secondary: 35J60, 58E11
Received: 12 January 2017
Revised: 3 June 2018
Accepted: 7 October 2018
Published: 21 May 2019
Jeffrey S. Case
Mathematics Department
Penn State University
University Park, PA
United States