TY - JOUR
T1 - The weighted σk-curvature of a smooth metric measure space
AU - Case, Jeffrey S.
N1 - Publisher Copyright:
© 2019 Mathematical Sciences Publishers.
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
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U2 - 10.2140/pjm.2019.299.339
DO - 10.2140/pjm.2019.299.339
M3 - Article
AN - SCOPUS:85067186551
SN - 0030-8730
VL - 299
SP - 339
EP - 399
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 2
ER -