Convergence of the calabi flow on toric varieties and related Kaehler manifolds

Let $X$ be a toric variety and $u$ be a normalized symplectic potential of the corresponding polytope $P$. Suppose that the Riemannian curvature is bounded by 1 and ${\int }_{\mathrm{\partial }P}ud\sigma <C_1,$ then there exists a constant $C_2$ depending only on $C_1$ and $P$ such that ${\max }_{P}u<C_2$. As an application, we show that if $(X,P)$ is analytic uniform $K$-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Riemannian curvature is uniformly bounded along the Calabi flow. Also we provide a proof of a conjecture of Donaldson. Finally, assuming that the curvature is bounded along the Calabi flow, our method would provide a proof of a conjecture due to Apostolov, Calderbank, Gauduchon and Tonnesen-Friedman.