Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow

Abstract

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 $K_X$ is nef. The abundance conjecture in algebraic geometry predicts that $K_X$ must be semi-ample when $X$ is a projective minimal model. We prove that if $K_X$ 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 $X_{can}$ 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 $X_{can}$, with uniformly bounded Ricci curvature away from the critical set of the pluricanonical map from $X$ to $X_{can}$.

References in corpus (6)

• The Kähler-Ricci flow on surfaces of positive Kodaira dimension
• Riemannian geometry of Kahler-Einstein currents
• Ricci flow and birational surgery
• Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow
• Entropy and heat kernel bounds on a Ricci flow background
• The local entropy along Ricci flow---Part A: the no-local-collapsing theorems

Cited by in corpus (2)

• Curvature Estimates for the Continuity Method
• On the improved no-local-collapsing theorem of Ricci flow