TY - JOUR

T1 - A notion of the weighted σk-curvature for manifolds with density

AU - Case, Jeffrey S.

N1 - Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/6/4

Y1 - 2016/6/4

N2 - We propose a natural definition of the weighted σk-curvature for a manifold with density; i.e. a triple (Mn, g, e-φdvol). This definition is intended to capture the key properties of the σk-curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointwise changes of the measure. We justify our definition through three main results. First, we show that shrinking gradient Ricci solitons are local extrema of the total weighted σk-curvature functionals when the weighted σk-curvature is variational. Second, we characterize the shrinking Gaussians as measures on Euclidean space in terms of the total weighted σk-curvature functionals. Third, we characterize when the weighted σk-curvature is variational. These results are all analogues of their conformal counterparts, and in the case k=1 recover some of the well-known properties of Perelman's W-functional.

AB - We propose a natural definition of the weighted σk-curvature for a manifold with density; i.e. a triple (Mn, g, e-φdvol). This definition is intended to capture the key properties of the σk-curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointwise changes of the measure. We justify our definition through three main results. First, we show that shrinking gradient Ricci solitons are local extrema of the total weighted σk-curvature functionals when the weighted σk-curvature is variational. Second, we characterize the shrinking Gaussians as measures on Euclidean space in terms of the total weighted σk-curvature functionals. Third, we characterize when the weighted σk-curvature is variational. These results are all analogues of their conformal counterparts, and in the case k=1 recover some of the well-known properties of Perelman's W-functional.

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U2 - 10.1016/j.aim.2016.03.010

DO - 10.1016/j.aim.2016.03.010

M3 - Article

AN - SCOPUS:84961830161

VL - 295

SP - 150

EP - 194

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -