Result 352, Differential geometry

A finite-time singularity of Calabi flow

Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth U(10)U(10)-invariant Kähler metric on CP10\mathbb{CP}^{10} whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.

Disproof or counterexample

The bigger picture

Why it matters

A geometric evolution intended to make curvature more uniform can instead become singular in finite time. The manuscript reports this failure on complex projective space, even when a constant-scalar-curvature metric is available in the same class.

What changes?

The manuscript constructs a smooth Kähler metric, a way of measuring lengths compatible with complex geometry, on complex projective space of complex dimension 10. It is invariant under the unitary group U(10) and belongs to the Fubini-Study class, the class of the standard metric. Under Calabi flow, which evolves such metrics to regularize curvature, its scalar curvature becomes unbounded in finite time. This counterexample contradicts Chen's conjecture that smooth Calabi flow always exists for all time.

What does that help mathematicians do?

The key distinction is between the existence of a preferred metric and the behavior of an evolution meant to reach it. This example shows that having a constant-scalar-curvature metric in the chosen class does not, by itself, prevent finite-time curvature blow-up. Researchers therefore cannot use that existence assumption alone to justify running Calabi flow smoothly for arbitrarily long times.

Are there practical applications?

The immediate value is foundational: the result identifies a limit of Calabi flow as a route to special metrics in complex geometry. Any approach relying on indefinite smooth evolution needs further hypotheses or a way to handle singularities. The construction does not say that every initial metric, or every dimension, exhibits this failure.

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.

Manuscript

A finite-time singularity of Calabi flow on projective space

September 24, 2026 54 pages

We construct a smooth Kähler metric in the Fubini–Study class of CP10\mathbb{CP}^{10} whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen's smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.

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Cite (BibTeX) 
@misc{OAI:A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026,
  author = {{OpenAI}},
  title = {{A finite-time singularity of Calabi flow on projective space}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf}{OAI:A-finite-time-singularity-of-Calabi-flow-on-projective-space-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 8 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.