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A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup
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Abstract
We disprove the scalar-curvature extension conjecture in its unrestricted all-dimensions form. In sufficiently high dimension, we construct a Ricci flow on a closed manifold whose scalar curvature remains uniformly bounded while full curvature diverges at a finite maximal time. In one fixed sufficiently high dimension, the examples have two-sided power-law curvature blowup with arbitrarily large exponents.
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Related prior literature mentioned by others
- Maxwell Stolarski, Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow (arXiv:1905.00087v1, 2019-04-30). AIProofWatch’s prior-work summary identifies these conical examples as preceding the manuscript’s additional bounded-scalar-curvature claim.
Possibly relevant literature (found by the agent, not checked)
- Shota Hamanaka, Type of finite time singularities of the Ricci flow with bounded scalar curvature (arXiv:2105.08250, 2021). Studies finite-time singularities of closed Ricci flows with uniformly bounded scalar curvature in dimensions 4 to 7; relevant to the paper's statement that its example lives only in sufficiently high dimension ($\dim = q+3\ge 13$).
- Nam Q. Le and Natasa Sesum, Remarks on the curvature behavior at the first singular time of the Ricci flow (Pacific J. Math. 255 (2012), 155–175). Shows that at Type II singularities with bounded scalar curvature suitable rescalings lead to a complete Ricci-flat limit; related to the paper's Type II example whose parabolic-scale model is a Ricci-flat cone.
- Reto Buzano and Gianmichele Di Matteo, A Local Singularity Analysis for the Ricci Flow and its Applications to Ricci Flows with Bounded Scalar Curvature (arXiv:2006.16227; Calc. Var. PDE 61 (2022), 65). Local Type I/Type II classification of singular points with applications to bounded scalar curvature; relevant to the local blowup rates in the paper's Corollary on two-sided power-law curvature bounds.
- Xianbin Huang and Wancheng Zhang, $L^2$ Curvature Bounds on Ricci flows with bounded scalar curvature up to dimension 7 (arXiv:2610.10575, 2026-10-05). Proves uniform $L^2$ curvature bounds for flows with bounded scalar curvature in dimensions $n\le 7$; gives a dimension range in which further a priori control holds, to compare with the paper's high-dimensional counterexample.
Digests and expositions
- Latent Seminar, Scalar curvature and Ricci-flow singularities: four-dimensional extension and higher-dimensional blowup (Latent Seminar, 2026-10-07). Explains the cap and parabolic scales and scalar-curvature barriers in this construction alongside the four-dimensional companion papers, explicitly without independently verifying the complete proofs. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 351
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026,
author = {{OpenAI}},
title = {{A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026/paper.pdf}{OAI:A-closed-Ricci-flow-with-bounded-scalar-curvature-and-finite-time-curvature-blowup-September-24-2026}},
year = {2026}
}Family 351: Scalar curvature and finite-time Ricci-flow singularities
Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.
Other papers in this family:
- Bounded scalar curvature and smooth extension of four-dimensional Ricci flow · September 24, 2026
- Path selection and an elliptic inequality on degenerating Ricci-flat trees · September 24, 2026