Paper · Nov 9, 2026
Curvature estimates for shrinking gradient Ricci solitons
As on arXiv
Abstract
Suppose is a complete gradient shrinking Ricci soliton with bounded scalar curvature. We prove that if , then its Riemannian curvature is bounded. The proof uses curvature blow-up, a local Ricci estimate, and the splitting given by the rescaled potential. An entropy estimate then allows us to apply the volume-ratio gap theorem for Ricci-flat four-manifolds.
- Submitted
- Nov 9, 2026
- Subjects
- math.DG
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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
- 6C.4The Yamabe Problem and Critical Exponents
Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.
- 9B.6Scalar Curvature and Topology
Positive scalar curvature, its obstructions, and the positive mass theorem.
- 6A.13Self-Similar Variables and Monotonicity
Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.