Journal
PACIFIC JOURNAL OF MATHEMATICS
Volume 299, Issue 2, Pages 339-399Publisher
PACIFIC JOURNAL MATHEMATICS
DOI: 10.2140/pjm.2019.299.339
Keywords
smooth metric measure space; sigma(k)-curvature; quasi-Einstein; weighted Einstein
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Funding
- NSF [DMS-1004394]
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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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