As on arXiv
Abstract
In this article, we prove an -regularity theorem for Perelman's reduced volume. We show that on a Ricci flow, if Perelman's reduced volume is close to , then the curvature radius at the base point cannot be too small.
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 2021, plus the canon.
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Deriving Perelman's entropy from Colding's monotonic volume
Topics Reduced volume · Shared terms “Perelman”
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Gromov-Hausdorff limits of the Chern-Ricci flow on smooth Hermitian minimal models of general type
Topics Reduced volume · Shared terms “Perelman reduced”
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Wrinkling of fluid deformable surfaces
Topics Reduced volume
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Pseudolocality and uniqueness of Ricci flow on almost Euclidean noncompact manifolds
Topics Ricci flow · Same authors Liang Cheng, Yongjia Zhang
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A gap theorem for metric solitons and its applications
Topics Ricci flow · Same author Yongjia Zhang · Shared terms “regularity theorem”, “curvature radius”
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On a parabolic curvature lower bound generalizing Ricci flows
Topics Reduced volume · Shared terms “Perelman”
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No breather theorems for the mean curvature flow
Same authors Liang Cheng, Yongjia Zhang
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Heat kernel on Ricci shrinker metric measure spaces
Topics Reduced volume
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
- 12A.5Reduced Distance and Reduced Volume
A space-time Bishop–Gromov theorem.
- 12A.6𝓛-Geodesics and the Variational Formulas
Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.
- 12A.7Monotonicity of the Reduced Volume
The 𝓛-exponential map and its Jacobian, the injectivity domain, the limit at τ = 0, barrier inequalities on complete flows, and min ℓ ≤ n/2, following Morgan and Tian's chapters 6 and 7.