Calabi flow on toric variety.

摘要

Let {(Mn, o + -1 ∂∂&phis;(t) > 0), 0 ≤ t T ≤ infinity} be a Calabi flow solution 6ft 6t=Rt Xiuxiong Chen and Weiyong He show that the Calabi flow can be extended if the Ricci curvature is bounded. They also show the long time existence of Calabi flow in Kahler surfaces under some extra conditions. We prove that in polarized toric manifold, the Calabi flow exists for all time if the following condition is satisfied: (1) The Ln2 -norms of Riemannian curvature are uniformly bounded in any finite time interval [0, T). (2) The Linfinity-norm of derivatives of Riemannian curvature at each time slice are bounded in [0, T) after rescaling by the Linfinity-norm of Riemannian curvature at that time slice. (3) The first derivative of scalar curvatures in Euclidean sense in the corresponding polytope are uniformly bounded in [0, T).