Cotton tensor and conformal deformations of three-dimensional Ricci flow

Abstract

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 $S^{1}$.