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Let  be a toric manifold with a Delzant polytope . We show that if (,) is analytic uniform -stable and the curvature is uniformly bounded along the Calabi flow, then the modified Calabi flow converges to an extremal metric exponentially fast. By assuming that the curvature is uniformly bounded along the Calabi flow, we prove a conjecture of Donaldson and a conjecture of Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman.

Convergence of the Calabi Flow on Toric Varieties and Related Kähler Manifolds