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Math-2307 · mathematics · Pde

Long-time existence and behavior of the Calabi flow

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
What is the behavior of $g_{j \overline k}(t)$ solving the Calabi flow, \[\frac \partial {\partial t} g_{j \overline k} = \partial_j \partial_{\overline k} R?\] Can long time existence be established for toric surfaces?

Formulation references: [S1], [S2].

Short English statement

What is the behavior of $g_{j \overline k}(t)$ solving the Calabi flow, \[\frac \partial {\partial t} g_{j \overline k} = \partial_j \partial_{\overline k} R?\] Can long time existence be established for toric surfaces?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2307-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2307

Finite-time Calabi-flow singularity: The counterexample supplies finite-time Calabi behavior on a toric projective tenfold; the bundled source also specifically asks long-time existence on toric surfaces, which is not resolved.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Long-time existence and behavior of the Calabi flow

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

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