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Bounded scalar curvature and smooth extension of four-dimensional Ricci flow

OpenAI · September 24, 2026

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Abstract

We prove that a smooth Ricci flow on a closed real four-manifold extends on the same manifold through every finite time at which its scalar curvature remains uniformly bounded. This resolves the scalar-curvature extension problem in dimension four.

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Possibly relevant literature (found by the agent, not checked)

Max Hallgren, Ricci Flow with Ricci Curvature and Volume Bounded Below (arXiv:2104.03386, 2021). For simply connected closed four-dimensional flows with Ricci curvature and volume bounded below, finite-time singularities are $C^0$ orbifolds and the flow continues by orbifold Ricci flow; a related four-dimensional continuation result, compared with the paper's claim of smooth continuation on $M$.
Wangjian Jian, Global $\varepsilon$-regularity for 4-dimensional Ricci flow with integral scalar curvature bound (arXiv:2008.06881; Pacific J. Math. 319 (2022), 333–342). Extends the Ge–Jiang global $\varepsilon$-regularity for four-dimensional flows with bounded scalar curvature; related to the paper's concentration and scale analysis in Section 4.
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). Part I of the local singularity analysis; the paper cites only Part II (arXiv:2606.12379) for scalar blowup at local Type I points.
Shota Hamanaka, Type of finite time singularities of the Ricci flow with bounded scalar curvature (arXiv:2105.08250, 2021). Studies blow-ups at locally Type I singularities of closed flows with bounded scalar curvature in dimensions 4 to 7; related to the special cases surveyed in the paper's introduction.
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). Extends the four-dimensional $L^2$ curvature and $L^4$ Ricci bounds of Bamler–Zhang, which the paper uses, to dimensions $n\le 7$.

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 outermost-bubble argument and its use of the static companion’s functional estimate alongside the higher-dimensional construction, explicitly without independently verifying the complete proofs. [family-level]

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 351

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