Scalar curvature and finite-time Ricci-flow singularities

3 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 3

A closed Ricci flow with bounded scalar curvature and finite-time curvature blowup

OpenAI · September 24, 2026

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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@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}
}

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

OpenAI · September 24, 2026

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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@misc{OAI:Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026,
  author = {{OpenAI}},
  title = {{Bounded scalar curvature and smooth extension of four-dimensional Ricci flow}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026/paper.pdf}{OAI:Bounded-scalar-curvature-and-smooth-extension-of-four-dimensional-Ricci-flow-September-24-2026}},
  year = {2026}
}

Path selection and an elliptic inequality on degenerating Ricci-flat trees

OpenAI · September 24, 2026

We prove a sequential elliptic inequality for the renormalized Einstein–Hilbert functional on a fixed finite tree of four-dimensional Ricci-flat spaces. As the joining lengths diverge and the scale-neutral weighted Ricci error M tends to zero, the functional satisfies \(E=o(M)\). The root end has decay exponent greater than one, and every joined quotient group is nontrivial. We also prove the multivariable path-selection theorem for asymptotic expansions used to obtain this estimate.

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@misc{OAI:Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026,
  author = {{OpenAI}},
  title = {{Path selection and an elliptic inequality on degenerating Ricci-flat trees}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026/paper.pdf}{OAI:Path-selection-and-an-elliptic-inequality-on-degenerating-Ricci-flat-trees-September-24-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.