Global existence and convergence of solutions of the Calabi flow on Einstein 4-manifolds

In this paper, firstly, we show the Bondi-mass type estimate of solutions o f Calabi flow on closed 4-manifolds. Secondly, in our applications, we obta in the long time existence on closed 4-manifolds. In particular, we are abl e to show the asymptotic convergence of a subsequence of solutions of the C alabi flow on closed Einstein 4-manifolds.