As on arXiv
Abstract
This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and Kähler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and Calabi-Yau structures as `canonical metrics' in generalized Riemannian and complex geometry. The generalized Ricci flow is introduced as a tool for constructing such metrics, and extensions of the fundamental Hamilton/Perelman regularity theory of Ricci flow are proved. These results are refined in the setting of generalized complex geometry, where the generalized Ricci flow is shown to preserve various integrability conditions, taking the form of pluriclosed flow and generalized Kähler-Ricci flow. This leads to global convergence results, and applications to complex geometry. A purely mathematical introduction to the physical idea of T-duality is given, and a discussion of its relationship to generalized Ricci flow.
- Submitted
- Aug 16, 2020
- Last revised
- Aug 28, 2020 (v2)
- Subjects
- math.DG, math.AP, math.CV
Comments: preliminary version, to appear in AMS University Lecture Series
Its neighbourhood
Connected papers
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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The generalized Kähler Calabi-Yau problem
Topics Generalized & G₂ flows, Ricci-flat & Calabi–Yau, Kähler–Ricci flow · Same author Jeffrey Streets · Shared terms “Yau structure”, “extension”
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Pluriclosed flow and the Hull-Strominger system
Topics Generalized & G₂ flows, Kähler geometry · Same authors Mario Garcia-Fernandez, Jeffrey Streets · Shared terms “pluriclosed flow”, “extension”, “relationship”
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Pluriclosed flow on Oeljeklaus-Toma manifolds
Topics Generalized & G₂ flows, Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “pluriclosed flow”
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Classification of generalized Kähler-Ricci solitons on complex surfaces
Topics Generalized & G₂ flows, Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “pluriclosed flow”
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Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces
Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”
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Deformation classes in generalized Kähler geometry
Topics Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”, “extension”, “preserves”
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The nondegenerate generalized Kähler Calabi-Yau problem
Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”
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Pluriclosed flow and the geometrization of complex surfaces
Topics Generalized & G₂ flows, Kähler geometry · Same author Jeffrey Streets · Shared terms “pluriclosed flow”, “fundamental aspects”
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.
Survey in the Generalized & G₂ flows guide: Garcia-Fernandez–Streets: *Generalized Ricci Flow*, the book-length account.
Survey in the Physics guide: Garcia-Fernandez–Streets: *Generalized Ricci Flow*, with its physics motivation.
In the guidebook
- 5A.6Where This Track Leads
The Kähler–Ricci flow, and when to come back to it.
- 5A.5Uniformization and the Two-Dimensional Ricci Flow
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.