Collapsing limits of the K\"ahler-Ricci flow and the continuity method

We consider the K\"ahler-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 K\"ahler-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 K\"ahler-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 K\"ahler 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 K\"ahler-Einstein current on a Riemann surface is compact.