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On the Collapsing Rate of the Kähler–Ricci Flow with Finite-Time Singularity

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We study the collapsing behavior of the Kähler–Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion \(X \xrightarrow{\pi} \varSigma\), where Σ is a Kähler manifold with \(\dim_{\mathbb{C}}\varSigma< \dim_{\mathbb{C}}X\). We give cohomological and curvature conditions under which the fibers π −1(z), z∈Σ collapse at the optimal rate \(\operatorname{diam}_{t} (\pi^{-1}(z)) \sim(T-t)^{1/2}\).

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Authors and Affiliations

  1. Department of Mathematics, Brown University, Providence, USA

    Frederick Tsz-Ho Fong

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  1. Frederick Tsz-Ho Fong
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Correspondence to Frederick Tsz-Ho Fong.

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Communicated by Bennett Chow.

This article forms a part of the author’s Ph.D. Thesis at Stanford University defended in May 2012. The author would like to thank his advisor Richard Schoen for his continuing encouragement and many inspiring ideas. He would also like to thank Simon Brendle and Yanir Rubinstein for many productive discussions and their interest in his work. He also thanks the anonymous referee for scrutinizing this work and for many helpful comments.

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Fong, F.TH. On the Collapsing Rate of the Kähler–Ricci Flow with Finite-Time Singularity. J Geom Anal 25, 1098–1107 (2015). https://doi.org/10.1007/s12220-013-9458-x

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  • DOI: https://doi.org/10.1007/s12220-013-9458-x

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