As on arXiv
Abstract
In this work, we obtain existence criteria for Chern-Ricci flows on noncompact manifolds. We generalize a result by Tossati-Wienkove on Chern-Ricci flows to noncompact manifolds and at the same time generalize a result for Kahler-Ricci flows by Lott-Zhang to Chern-Ricci flows. Using the existence results, we prove that any complete noncollapsed Kahler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kahler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume. Combining this result with the result of Chau-Tam, we give another proof that a complete noncompact Kahler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to the complex Euclidean space. This last result has already been proved by Gang Liu recently using other methods. This last result is partial confirmation of a uniformization conjecture of Yau.
- Submitted
- Aug 1, 2017
- Last revised
- Aug 17, 2017 (v2)
- Subjects
- math.DG
Comments: some results added; printing mistakes corrected
Its neighbourhood
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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.
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On existence and curvature estimates of Ricci flow
Topics Surfaces, Kähler–Ricci flow, (Non)collapsing · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “nonnegative bisectional”, “bisectional curvature”, “noncompact Kahler”
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Kähler manifolds with almost non-negative curvature
Topics Kähler–Ricci flow, (Non)collapsing, Boundaries & non-compact · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “bisectional curvature”, “maximal volume”, “complex manifold”
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Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics
Topics Generalized & G₂ flows, Boundaries & non-compact, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “Chern Ricci”, “complex manifold”, “Lott Zhang”
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Longtime existence of Kähler Ricci flow and holomorphic sectional curvature
Topics Kähler–Ricci flow, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “existence criteria”, “Yau”
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Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds
Topics Surfaces, Kähler–Ricci flow, Boundaries & non-compact · Shared terms “complete Kähler”, “volume growth”, “nonnegative bisectional”
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Hermitian manifolds with quasi-negative curvature
Topics Generalized & G₂ flows · Same author Man-Chun Lee · Shared terms “Chern Ricci”, “bisectional curvature”, “initial metric”
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An almost complex Chern-Ricci flow
Topics Generalized & G₂ flows, Kähler–Ricci flow, Kähler geometry · Shared terms “Chern Ricci”, “complex manifold”, “initial metric”
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Ancient solutions to the Kähler Ricci flow
Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Shared terms “nonnegative bisectional”, “bisectional curvature”, “noncollapsed”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 11A.12Ricci Flow on Surfaces
Uniformization by Ricci flow, and Hamilton’s entropy.
- 5A.6Where This Track Leads
The Kähler–Ricci flow, and when to come back to it.