Collapsing limits of the Kähler–Ricci flow and the continuity method

Abstract

We consider the Kahler–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 Kahler–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 Kahler–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 Kahler 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 Kahler–Einstein current on a Riemann surface is compact.