In this paper, we study the Calabi flow on a polarized Kähler manifold and some related problems. We first give a precise statement on the short time existence of the Calabi flow for any c3,α(M) initial Kähler potential. As an application, we prove a stability result: any metric near a constant scalar curvature Kähler (CscK) metric will flow to a nearby CscK metric exponentially fast. Secondly, we prove that a compactness theorem in the space of the Kähler metrics given uniform Ricci bound and potential bound. As an application, we prove the Calabi flow can be extended once Ricci curvature stays uniformly bounded. Lastly, we prove a removing-singularity result about a weak constant scalar curvature metric in a punctured disc.


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pp. 539-570
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