As on arXiv
Abstract
This paper defines a parabolic frequency for solutions of the heat equation on a Ricci flow and proves it's monotonicity along the flow. Frequency monotonicity is known to have many useful consequences; here it is shown to provide a simple proof of backwards uniqueness. For solutions of more general parabolic equations on a Ricci flow, this paper provides bounds on the derivative of the frequency, which similarly imply backwards uniqueness.
Comments: 14 pages; final version published in IMRN
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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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Parabolic frequency for the mean curvature flow
Topics Stability & uniqueness · Same author Julius Baldauf · Shared terms “parabolic frequency”, “backwards uniqueness”, “imply backwards”
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Parabolic frequency monotonicity for two nonlinear equations under Ricci flow
Topics Ricci flow · Shared terms “parabolic frequency”, “frequency monotonicity”, “parabolic equations”
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Parabolic frequency monotonicity on Ricci flow and Ricci-harmonic flow with bounded curvatures
Topics Ricci flow · Shared terms “parabolic frequency”, “frequency monotonicity”, “heat equation”
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Parabolic frequency monotonicity on the conformal Ricci flow
Topics Ricci flow · Shared terms “parabolic frequency”, “frequency monotonicity”, “heat equation”
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Gradient estimates and parabolic frequency monotonicity for positive solutions of the heat equation under generalized Ricci flow
Topics Ricci flow · Shared terms “parabolic frequency”, “frequency monotonicity”, “heat equation”
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Matrix Li-Yau-Hamilton estimates under Ricci Flow and parabolic frequency
Topics Ricci flow · Shared terms “parabolic frequency”, “heat equation”, “backward”
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Ricci Flow on ALF manifolds
Topics Stability & uniqueness · Same author Dain Kim
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Harmonic spinors in the Ricci flow
Topics Ricci flow · Same author Julius Baldauf
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.