Osaka Journal of Mathematics

Cotton tensor and conformal deformations of three-dimensional Ricci flow

Yoshihiro Umehara

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In this paper, we study the deformation of the three-dimensional conformal structures by the Ricci flow. We drive the evolution equation of the Cotton--York tensor and the $L^{1}$-norm of it under the Ricci flow. In particular, we investigate the behavior of the $L^{1}$-norm of the Cotton--York tensor under the Ricci flow on three-dimensional simply-connected Riemannian homogeneous spaces which admit compact quotients. For a non-homogeneous case, we also investigate the behavior of the $L^{1}$-norm for the product metric of the Rosenau solution for the Ricci flow on $S^{2}$ and the standard metric of $S^{1}$.

Article information

Osaka J. Math., Volume 53, Number 2 (2016), 515-534.

First available in Project Euclid: 27 April 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A30: Conformal differential geometry 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)


Umehara, Yoshihiro. Cotton tensor and conformal deformations of three-dimensional Ricci flow. Osaka J. Math. 53 (2016), no. 2, 515--534.

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