Result 351, Differential geometry

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.

Proof

The bigger picture

Why it matters

Ricci flow evolves a space's geometry according to its curvature. These manuscripts claim that one numerical summary detects every finite-time breakdown on closed four-dimensional spaces, but can miss breakdowns in sufficiently high dimensions.

What changes?

The four-dimensional manuscript reports smooth extension on the same manifold when scalar curvature, a single-number curvature summary at each point, stays uniformly bounded up to a finite time. It assumes a smooth Ricci flow on a closed real four-manifold: compact and without boundary. The counterexample manuscript constructs closed flows in sufficiently high dimension with bounded scalar curvature but divergent full curvature at a finite maximal time. In one fixed such dimension, blowup has two-sided power-law bounds with arbitrarily large exponents.

What does that help mathematicians do?

In four dimensions, the claimed extension theorem rules out a specific hidden failure: increasingly extreme full curvature cannot end a smooth closed flow while scalar curvature remains uniformly controlled. Thus a genuine finite-time singularity must also lose that scalar bound. The high-dimensional examples show why this diagnostic cannot be adopted universally, even when only a single fixed dimension is considered.

Are there practical applications?

The immediate value is foundational, in understanding singularities of evolving geometries. The results identify when controlling one curvature quantity guarantees continued smooth evolution, and when that strategy fails. They therefore distinguish a sufficient target for four-dimensional extension arguments from an insufficient one in higher dimensions, rather than supplying a practical computational method.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

3 manuscripts

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

September 24, 2026 20 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 24, 2026 58 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 24, 2026 40 pages

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)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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.