Diameter estimates for long-time solutions of the Kähler–Ricci flow
Diameter estimates for long-time solutions of the Kähler–Ricci flow
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DOI：
10.1007/s00039-022-00620-9
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发表时间：
2022-10
影响因子：
2.2
通讯作者：
Wangjian Jian;Jian Song
中科院分区：
文献类型：
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作者：
Wangjian Jian;Jian Song
众所周知，凯勒流形 X 上的凯勒-里奇流承认长期解当且仅当 X 是最小模型，即规范线束是 nef。代数几何中的丰度猜想预测，当 X 是射影极小模型时，必定是半丰度。我们证明，如果是半充足的，那么对于归一化 Kähler-Ricci 流的长期解，直径是一致有界的。我们对 Kähler-Ricci 流的长期解的直径估计与 Song 和 Tian (Am J Math 138(3):683–695, 2016) 中的标量曲率估计相结合，是佩雷尔曼直径和法诺流形上短时解的标量曲率估计的自然扩展。作为一个应用，最小三重上的归一化 Kähler-Ricci 流总是以 Gromov-Hausdorff 拓扑顺序收敛到与其规范模型同胚的紧度量空间。
It is well known that the Kähler–Ricci flow on a Kähler manifoldXadmits a long-time solution if and only ifXis a minimal model, i.e., the canonical line bundleis nef. The abundance conjecture in algebraic geometry predicts thatmust be semi-ample whenXis a projective minimal model. We prove that ifis 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 threefoldXalways converges sequentially in Gromov–Hausdorff topology to a compact metric space homeomorphic to its canonical model.