A finite-time singularity of Calabi flow

Claims a smooth starting metric on complex projective 10-space whose Calabi flow breaks down in finite time.

1 paper September 24, 2026 Explained by AI

Calabi flow is an equation that slowly reshapes a curved complex space, aiming to make its curvature even. Chen conjectured that it runs forever from any smooth start. The paper claims a smooth start on complex projective 10-space where the curvature becomes infinite at one point in finite time. The space even carries a perfectly even metric, so the failure is not caused by the space itself. There is no Lean check.

First step

1982

Eugenio Calabi, who introduced the flow

Significance

Major

It claims to disprove a standing conjecture of Chen on smooth long-time existence for Calabi flow.

Trust

Claimed, not checked

Field

Differential geometry

1 paper

The explanation

Start with complex projective space of dimension 10, CP¹⁰. It is the set of complex lines through the origin in an 11-dimensional complex space. It carries a standard symmetric metric, the Fubini–Study metric. Scalar curvature is one number at each point that averages all the bending there. With the paper's normalization, Fubini–Study has scalar curvature 110 at every point. That is 10 × 11, the dimension times one more.

A Kähler metric is a way to measure lengths on a complex space that fits its complex structure. Calabi flow changes such a metric over time. Where scalar curvature is above average it pushes one way, and below average the other. Calabi introduced it to find the most even metric in a familySource 1. Chen conjectured that the flow runs smoothly forever from any smooth startSource 2. On surfaces of complex dimension one this was known: Chruściel proved the flow lasts forever and settles downSource 3. In higher dimensions the flow was known to continue while scalar curvature stays boundedSource 4.

The paper claims a no. It builds a smooth starting metric on CP¹⁰, in the same family as Fubini–StudySource 5. The flow stops at a finite time T*. At one point p the scalar curvature grows like a constant times 1/√(T* − t)Source 5. So when the time left falls from 0.01 to 0.0001, the curvature there grows about tenfold. Because the family contains the even Fubini–Study metric, the failure happens even where a perfect target existsSource 6. Taking products with a fixed even factor gives the same breakdown in every complex dimension of at least 10Source 7.

How does the construction work? The metric is symmetric under a large group, U(10), so the flow reduces to one equation in one variable. At p the scalar curvature equals 110 times one minus a single value H(t,0)Source 8. The paper builds an exact shrinking solution of a simpler flat model equationSource 9. A computer calculation with exact arithmetic checks the matching inequalities for that solutionSource 10. The hard step is the correction on the compact space. It must stay smaller than the main term at p, so the blow-up survivesSource 11. The starting metric is chosen at the end, as a time slice of the solutionSource 12.

How sure is this? There is no Lean check. A computer proof checker has not looked at the analysis. The profile step relies on a computer certificate that the paper includes. No specialist review of the paper has been published.

What it doesn't show. This does not show that weak versions of the flow fail, and it says nothing about complex dimensions below ten. It has no Lean check.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if the construction holds: it would disprove Chen's long-time existence conjecture for Calabi flow. There is no Lean check, part of the proof is a computer certificate, and no specialist review is published.

Words used

Kähler metric: A way to measure lengths on a complex space that fits its complex structure.
Calabi flow: An equation that changes a Kähler metric over time to even out its scalar curvature.
Scalar curvature: One number at each point that averages all the ways the space bends there.
Complex projective space: The space of complex lines through the origin; CP¹⁰ has complex dimension 10.
Fubini–Study metric: The standard, fully symmetric metric on complex projective space; its scalar curvature is constant.
Kähler class: The family of Kähler metrics that differ by a smooth potential function; the flow stays inside one class.
Finite-time singularity: A time at which the flow cannot continue smoothly because curvature becomes infinite.

The original paper

A finite-time singularity of Calabi flow on projective space (PDF, opens in a new tab)
September 24, 2026

All quoted lines (12)

Calabi introduced this flow to seek canonical metrics within a K\"ahler class.
A finite-time singularity of Calabi flow on projective space Back to text
Chen's smooth long-time existence conjecture asserts that the flow exists for all positive times from arbitrary smooth K\"ahler initial data; see.
A finite-time singularity of Calabi flow on projective space Back to text
On compact Riemann surfaces, Chru\'sciel proved global forward existence and convergence to a constant-curvature metric.
A finite-time singularity of Calabi flow on projective space Back to text
Chen and He proved extension under a uniform Ricci-curvature bound, and Chen and Cheng established extension under a uniform scalar-curvature bound.
A finite-time singularity of Calabi flow on projective space Back to text
There is a smooth $U(10)$ -invariant K\"ahler form $\omega_0$ on $\CP^{10}$ , in the class $[\omega_{\mathrm{FS}}]$ , whose maximal smooth Calabi flow has existence interval $[0,T_*)$ with $0<T_*<\infty$ .
A finite-time singularity of Calabi flow on projective space Back to text
Thus the example also answers negatively the smooth global-existence question for classes containing a constant-scalar-curvature K\"ahler metric, emphasized by Chen and Cheng.
A finite-time singularity of Calabi flow on projective space Back to text
In particular, finite-time smooth Calabi-flow singularities occur in every complex dimension $n\geq10$ , on $\CP^{10}\times\CP^{n-10}$ , in a class containing a K\"ahler–Einstein metric.
A finite-time singularity of Calabi flow on projective space Back to text
At the fixed affine origin $p$ , the scalar-curvature identity is $S(t,p)=110\bigl(1-H(t,0)\bigr).$
A finite-time singularity of Calabi flow on projective space Back to text
Section … constructs $G$ so that $F$ solves the scale-invariant flat equation exactly.
A finite-time singularity of Calabi flow on projective space Back to text
A finite exact-arithmetic calculation, with analytic enclosure and nonlinear error bounds, certifies the matching inequalities.
A finite-time singularity of Calabi flow on projective space Back to text
The correction is therefore smaller than the leading term at the origin, preserving the desired scalar-curvature divergence.
A finite-time singularity of Calabi flow on projective space Back to text
A smooth interior time slice of this solution will be the initial metric; no initial metric needs to be fixed before the construction.
A finite-time singularity of Calabi flow on projective space Back to text

Other sources

openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

361Failure of integer-degree harmonic dimension comparisonDifferential geometry
339Katok's entropy rigidity conjectureDifferential geometry
338Yau's uniformization conjectureDifferential geometry
336Spectral scalar curvature, Urysohn width, and macroscopic dimensionDifferential geometry
335Gromov’s integral scalar-curvature bound for simplicial volumeDifferential geometry

Browse Differential geometry Curvature and geometric flows Open since the 1980s Claimed, not yet checked