Toric Surfaces, K-Stability and Calabi Flow

Let $X$ be a toric surface and $u$ be a normalized symplectic potential on the corresponding polygon $P$. Suppose that the Riemannian curvature is bounded by a constant $C_1$ and ${\int }_{\mathrm{\partial }P}ud\sigma <C_2,$ then there exists a constant $C_3$ depending only on $C_1,C_2$ and $P$ such that the diameter of $X$ is bounded by $C_3$. Moreoever, we can show that there is a constant $M>0$ depending only on $C_1,C_2$ and $P$ such that Donaldson’s $M$-condition holds for $u$. As an application, we show that if $(X,P)$ is (analytic) relative $K$-stable, then the modified Calabi flow converges to an extremal metric exponentially fast by assuming that the Calabi flow exists for all time and the Riemannian curvature is uniformly bounded along the Calabi flow.