As on arXiv
Abstract
This paper studies the Ricci flow on closed manifolds admitting harmonic spinors. It is shown that Perelman's Ricci flow entropy can be expressed in terms of the energy of harmonic spinors in all dimensions, and in four dimensions, in terms of the energy of Seiberg-Witten monopoles. Consequently, Ricci flow is the gradient flow of these energies. The proof relies on a weighted version of the monopole equations, introduced here. Further, a sharp parabolic Hitchin-Thorpe inequality for simply-connected, spin 4-manifolds is proven. From this, it follows that the normalized Ricci flow on any exotic K3 surface must become singular.
Comments: 19 pages
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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 2021, plus the canon.
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On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons
Topics 4-manifolds · Shared terms “Thorpe inequality”, “Hitchin Thorpe”
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Spinors and mass on weighted manifolds
Topics Ricci flow · Same author Julius Baldauf · Shared terms “Witten”, “gradient flow”, “Perelman”
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Parallel spinor flows on three-dimensional Cauchy hypersurfaces
Topics Ricci-flat & Calabi–Yau, 4-manifolds · Shared terms “simply connected”, “admitting”
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The spinorial energy for asymptotically Euclidean Ricci flow
Topics Ricci flow · Same author Julius Baldauf · Shared terms “gradient flow”, “Perelman”
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Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus
Topics Ricci-flat & Calabi–Yau, 4-manifolds · Shared terms “gradient flow”
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On a dichotomy of the curvature decay of steady Ricci soliton
Topics Ricci-flat & Calabi–Yau, 4-manifolds
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Classification of Gradient Ricci solitons with harmonic Weyl curvature
Topics Ricci-flat & Calabi–Yau, 4-manifolds
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Diameter estimates and Hitchin-Thorpe inequality for four-dimensional compact Quasi-Einstein manifolds
Topics 4-manifolds · Shared terms “Thorpe inequality”, “Hitchin Thorpe”
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.
- 10A.12Curve Shortening and Minimal Discs
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.
- 11A.1The Equation and Its First Solutions
A heat equation for the metric: spheres, cylinders and scaling.