  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_{\partial P} u ~ d \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.

Differential Geometry