On the existence of extremal metrics for $L^2$-norm of scalar curvature on closed 3-manifolds

Abstract

In this paper, based on Bochner formula, mass decay estimates and elliptic Moser iteration, we show the global existence and asymptotic convergence of a subsequence of solutions of Calabi flow on some closed 3-manifolds, and then the existence of extermal metrics of $L^{2}$-norm of scalar curvature functional on a fixed conformal class is claimed. In particular, we may re-solve part of the Yamabe conjecture on closed 3-manifolds.