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 \(X_{\text {can}}\).

众所周知，Kähler 流形X上的 Kähler-Ricci 流当且仅当X是一个极小模型，即规范线丛\(K_X\)是 nef时才承认长期解。代数几何中的丰度猜想预测，当X是射影最小模型时， \(K_X\)一定是半充足的。我们证明如果\(K_X\)是半充足的，那么对于归一化 Kähler-Ricci 流的长时间解，直径是一致有界的。我们的直径估计与 Song 和 Tian (Am J Math 138(3):683–695, 2016) 中关于 Kähler-Ricci 流的长期解的标量曲率估计相结合是 Perelman 的直径和标量曲率估计的自然扩展Fano流形的短期解决方案。作为一个应用，最小三重X上的归一化 Kähler-Ricci 流总是在 Gromov-Hausdorff 拓扑中顺序收敛到与其规范模型\(X_{\text {can}}\)同胚的紧凑度量空间。