Chern-ricci flows on noncompact complex manifolds

Man Chun Lee, Luen Fai Tam

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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 languageEnglish (US)
Pages (from-to)529-564
Number of pages36
JournalJournal of Differential Geometry
Volume115
Issue number3
DOIs
StatePublished - Jul 2020

Keywords

  • Chern-Ricci flow
  • Holomorphic bisectional curvature
  • Kähler manifold
  • Uniformization

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

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