As on arXiv
Abstract
We first show that any -dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy for some positive constant . Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for -dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a -dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by Chodosh 14 in dimension three and show that any -dimensional gradient Ricci expander asymptotic to is rotationally symmetric, where is a constant and is the standard metric on with constant curvature .
Comments: 34 pages
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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Lower bounds for the scalar curvatures of Ricci flow singularity models
Topics Ricci-flat & Calabi–Yau, Singularities, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “singularity models”, “flat steady”, “lower bound”
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Volume growth estimates of gradient Ricci solitons
Topics Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “gradient Ricci”, “expanding gradient”
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On a dichotomy of the curvature decay of steady Ricci soliton
Topics Ricci-flat & Calabi–Yau, Ricci solitons · Same author Pak-Yeung Chan · Shared terms “flat steady”, “proper potential”, “steady gradient”
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Curvature pinching of asymptotically conical gradient expanding Ricci solitons
Topics Curvature pinching, 3-manifolds, Ricci solitons · Shared terms “Hamilton Ivey”, “Ricci expander”, “dimensional gradient”
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An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons
Topics Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “steady gradient”, “gradient Ricci”
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Dimension Reduction for Positively Curved Steady Solitons
Topics Singularities, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “steady gradient”, “gradient Ricci”
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On noncollapsed -limit metric solitons
Topics Singularities, Ricci solitons · Same authors Pak-Yeung Chan, Zilu Ma, Yongjia Zhang · Shared terms “singularity models”, “lower bound”
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Curvature estimates for steady and expanding solitons in higher dimensions
Topics Ricci solitons · Same author Pak-Yeung Chan · Shared terms “proper potential”, “curvature operator”, “Ricci expander”
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
- 12B.6Bounded Curvature at Bounded Distance
The heart of the canonical neighbourhood proof: why curvature at bounded rescaled distance from a point of high curvature stays bounded, via incomplete limits, cones and Hamilton–Ivey pinching (Morgan–Tian, chapter 10).
- 6A.9Reaction–Diffusion: Fronts, Patterns and Pinching
Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.
- 11A.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.