Abstract
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 non-collapsed 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 Cn. 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].
Original language | English (US) |
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Pages (from-to) | 529-564 |
Number of pages | 36 |
Journal | Journal of Differential Geometry |
Volume | 115 |
Issue number | 3 |
DOIs | |
State | Published - Jul 2020 |
Funding
Mathematics Subject Classification. Primary 32Q15, Secondary 53C44. Key words and phrases. Chern-Ricci flow, Kähler manifold, holomorphic bisectional curvature, uniformization. †Research partially supported by Hong Kong RGC General Research Fund #CUHK 14301517. Received September, 2017.
Keywords
- Chern-Ricci flow
- Holomorphic bisectional curvature
- Kähler manifold
- Uniformization
ASJC Scopus subject areas
- Analysis
- Algebra and Number Theory
- Geometry and Topology