Diameter estimates for long-time solutions of the Kähler–Ricci flow

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 Song and Tian (Am J Math 138(3):683–695, 2016) 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. 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 .