On the Calabi Flow

Abstract

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 c 3,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.

The pseudo-Calabi flow

“…So we obtain Theorem 5.7 (Chen-He [17]). Suppose that the Ricci curvature Ric j is uniformly bounded above, that is,…”

“…[17]), we have Theorem 1.2. The L y bound of the Ricci curvature is the obstruction to an extension of the pseudo-Calabi flow.…”

“…A critical point of this Calabi flow is precisely a cscK metric. This flow has been actively studied in recent years; cf., e.g., [12], [16], [17], [18], [24], [58]. However, the remaining technical di‰culties are still daunting simply because it is a 4th order flow.…”

A splitting theorem on toric manifolds

“…Note that by Chen and He [7], the Calabi flow exists for a short time starting from any C 3,α relative Kähler potential. Thus for a smooth symplectic potential u, the Calabi flow starting from u also exists for a short time.…”

Convergence of metrics under self-dual Weyl tensor and scalar curvature bounds

“…The motivation here also comes from the Calabi flow. In estimating whether the solution of Calabi flow exists and convergence, one crucial estimate which still lacking is one showing that the space of all kähler metrics ω in a fixed cohomoplogy class Ω, and for which the scalar curvature |S(ω)| ω is bounded, is compact in the C 1,α Cheeger -Gromov topology, see [Calabi82] and [ChHe08]. Then there exist a subsequence {j} ⊂ {i} such that (M, g j ) converge to a compact multi -fold (M ∞ , g ∞ ) in the Gromov -Hausdorff topology.…”