Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: econ
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The categories of this paper are: q-alg