Collapsing limits of the K\"ahler-Ricci flow and the continuity method
Abstract
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.
Keywords
Cite
@article{arxiv.1705.01434,
title = {Collapsing limits of the K\"ahler-Ricci flow and the continuity method},
author = {Yashan Zhang},
journal= {arXiv preprint arXiv:1705.01434},
year = {2018}
}
Comments
V3, some incorrect sentences removed, typos fixed, references updated; accepted by Mathematische Annalen