Collapsing limits of the Kähler–Ricci flow and the continuity method
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We consider the Kähler–Ricci flow on certain Calabi–Yau fibration, which is a Calabi–Yau fibration with one dimensional base or a product of two Calabi–Yau fibrations with one dimensional bases. Assume the Kähler–Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov–Hausdorff topology to the metric completion of the regular part of generalized Kähler–Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi–Yau fibrations are also obtained. Moreover, we show the continuity method starting from a suitable Kähler metric on the total space of a Fano fibration with one dimensional base converges in Gromov–Hausdorff topology to a compact metric on the base. During the proof, we show the metric completion of the regular part of a generalized Kähler–Einstein current on a Riemann surface is compact.
Mathematics Subject Classification53C44 53C55
The author is grateful to Professor Huai-Dong Cao for constant encouragement and support and Professor Valentino Tosatti for his crucial help during this work, valuable suggestions on a previous draft and constant encouragement. He also thanks Professor Hans-Joachim Hein for communications which motivate Remark 3, Professor Chengjie Yu for useful comments on a previous draft, Professor Zhenlei Zhang for collaboration and encouragement and Peng Zhou for kind help. This work was carried out while the author was visiting Department of Mathematics at Northwestern University, which he would like to thank for the hospitality. The author is grateful to the referee and editor for their careful reading and very useful suggestions and corrections, which help to improve this paper. Very recently, Fu et al.  made a big progress on studying the geometry of the continuity method. They proved that the diameter of \(\omega (t)\) solving from the continuity method (1.8) or (1.11) is uniformly bounded. Their result in particular gives an alternative proof for the diameter upper bound of the continuity method involved in proofs of Theorems 4, 5 and 7.
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