Paper · Oct 6, 2025
Curvature pinching of asymptotically conical gradient expanding Ricci solitons
As on arXiv
Abstract
In this paper, we investigate curvature pinching phenomena in complete non-compact asymptotically conical gradient expanding Ricci solitons and establish several Hamilton-Ivey type curvature pinching estimates. These results are parallel to those known for shrinking and steady Ricci solitons. In particular, we prove a three-dimensional Hamilton-Ivey type curvature pinching theorem: any three-dimensional non-compact gradient Ricci expander, which is asymptotic to a cone with positive scalar curvature, must have positive sectional curvature. Furthermore, we formulate a general method and apply it to obtain analogues of several additional known generalized Hamilton-Ivey type curvature pinching results for ancient solutions. Among these is a curvature pinching estimate for four-dimensional asymptotically conical Ricci expanders with uniformly positive isotropic curvature, analogous to a result for four-dimensional gradient steady solitons due to Brendle [8].
Comments: 29 pages; minor updates
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A note on the classification of four-dimensional gradient steady and expanding Ricci solitons
Topics 4-manifolds, Boundaries & non-compact, Ricci solitons · Same authors Huai-Dong Cao, Junming Xie · Shared terms “expanding Ricci”, “gradient steady”, “dimensional gradient”
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Four-dimensional gradient Ricci solitons with (half) nonnegative isotropic curvature
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Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature
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Complete gradient expanding Ricci solitons with finite asymptotic scalar curvature ratio
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Curvature estimates for four-dimensional complete gradient expanding Ricci solitons
Topics 4-manifolds, Scalar curvature, Ricci solitons · Same author Huai-Dong Cao · Shared terms “gradient expanding”, “expanding Ricci”, “four dimensional”
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On curvature estimates for four-dimensional gradient Ricci solitons
Topics 4-manifolds, Ricci solitons · Same author Huai-Dong Cao · Shared terms “dimensional gradient”, “four dimensional”, “gradient steady”
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Hamilton-Ivey estimates for gradient Ricci solitons
Topics Curvature pinching, 3-manifolds, Ricci solitons · Shared terms “Hamilton Ivey”, “Ricci expander”, “dimensional gradient”
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On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions
Topics Curvature pinching, Sphere theorems, 4-manifolds · Shared terms “positive isotropic”, “isotropic curvature”, “pinching estimate”
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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.2The Asymptotic Soliton and Three-Dimensional Shrinkers
Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.
- 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.