Extremal Metrics for Quadratic Functional of Scalar Curvature on Closed 3-Manifolds

摘要

In this paper, we first show the global existence of the three-dimensional Calabi flow on any closed 3-manifold with an arbitrary background metric g0. Second, we show the asymptotic convergence of a subsequence of solutions of the Calabi flow on a closed 3-manifold with Yamabe constant Q < 0 or Q = 0 and Q > 0, up to conformal transformations. With its application, we prove the existence of extremal metrics for quadratic functional of scalar curvature on a closed 3-manifold which is served as an extension of the Yamabe problem on closed manifolds. Moreover, the existence of extremal metrics on complete noncompact 3-manifolds will discuss elsewhere.