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July 2020 Chern–Ricci flows on noncompact complex manifolds
Man-Chun Lee, Luen-Fai Tam
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J. Differential Geom. 115(3): 529-564 (July 2020). DOI: 10.4310/jdg/1594260018


In this work, we obtain existence criteria for Chern–Ricci flows on noncompact manifolds. We generalize a result by Tossati–Wienkove [37] on Chern-Ricci flows to noncompact manifolds and a result for Kähler–Ricci flows by Lott–Zhang [21] to Chern–Ricci flows. Using the existence results, we prove that any complete noncollapsed Kähler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kähler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume growth. Combining this result with [3], we give another proof that a complete noncompact Kähler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to $\mathbb{C}^n$. This last result has already been proved by Liu [20] recently using other methods. This last result is a partial confirmation of a uniformization conjecture of Yau [41].

Funding Statement

Research partially supported by Hong Kong RGC General Research Fund #CUHK 14301517.


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Man-Chun Lee. Luen-Fai Tam. "Chern–Ricci flows on noncompact complex manifolds." J. Differential Geom. 115 (3) 529 - 564, July 2020.


Received: 1 September 2017; Published: July 2020
First available in Project Euclid: 9 July 2020

zbMATH: 07225030
MathSciNet: MR4120818
Digital Object Identifier: 10.4310/jdg/1594260018

Primary: 32Q15
Secondary: 53C44

Keywords: Chern–Ricci flow , holomorphic bisectional curvature , Kähler manifold , uniformization

Rights: Copyright © 2020 Lehigh University


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Vol.115 • No. 3 • July 2020
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