It is well known that the Kähler-Ricci flow on a Kähler manifold X admits a long-time solution if and only if X is a minimal model, i.e., the canonical line bundle KX is nef. The abundance conjecture in algebraic geometry predicts that KX must be semi-ample when X is a projective minimal model. We prove that if KX is semi-ample, then the diameter is uniformly bounded for long-time solutions of the normalized Kähler-Ricci flow. Our diameter estimate combined with the scalar curvature estimate in [34] for long-time solutions of the Kähler-Ricci flow are natural extensions of Perelman's diameter and scalar curvature estimates for short-time solutions on Fano manifolds. We further prove that along the normalized Kähler-Ricci flow, the Ricci curvature is uniformly bounded away from singular fibres of X over its unique algebraic canonical model Xcan if the Kodaira dimension of X is one. As an application, the normalized Kähler-Ricci flow on a minimal threefold X always converges sequentially in Gromov-Hausdorff topology to a compact metric space homeomorphic to its canonical model Xcan, with uniformly bounded Ricci curvature away from the critical set of the pluricanonical map from X to Xcan.
We establish a general "boundedness implies convergence" principle for a family of evolving Riemannian metrics. We then apply this principle to collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows on torus fibered minimal models to obtain convergence results.2, 419-433
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