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Math-405-A2 · revision r1 · on Math-405

OpenAI claim summary for Math-405

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Finite-time Calabi-flow singularity: The initial smooth form is in the Fubini-Study class on projective ten-space, which contains a Kahler-Einstein and hence constant-scalar-curvature metric. Its finite maximal smooth time contradicts the required global existence, already enough to negate the conjunction with smooth convergence.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Long-Time Existence of the Calabi Flow on Compact Kaehler Manifolds
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Long-Time Existence of the Calabi Flow on Compact Kaehler Manifolds

OpenAI's claim

Finite-time Calabi-flow singularity: The manuscript claims a smooth Kähler metric in the standard class on $\mathbb {CP}^{10}$ whose scalar curvature becomes unbounded during Calabi evolution before a finite time. Thus the failure can occur in a class that also contains a constant-scalar-curvature metric.

Mathematical statement of the claimed result

Mathematical claim 1.
The claimed initial Kähler form $\omega_0$ lies in the Fubini--Study class on $\mathbb{CP}^{10}$, is smooth, and is invariant under $U(10)$ whose maximal smooth Calabi flow has finite lifespan $0<T_*<\infty$. At some fixed point $p$ and with $a>0$, \[S(\omega(t))(p)\sim a(T_*-t)^{-1/2}\quad(t\uparrow T_*).\] Here the flow is $\partial_t\omega=i\partial\bar\partial S(\omega)$, and $S(\omega)=\operatorname{tr}_{\omega}\operatorname{Ric}(\omega)$, using the complex Kähler trace convention. This refutes universal smooth long-time existence.

Outline of the argument

Finite-time Calabi-flow singularity: A radial momentum ansatz reduces the geometric flow to a fourth-order scalar equation with smooth endpoint conditions. An exact shrinking solution of the flat model is constructed first. A correction on the compact projective manifold preserves the shrinking mechanism and the positivity of the metric, giving the finite-time curvature singularity.

Formal verification

Finite-time Calabi-flow singularity: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources