Chen's conjecture on global existence of the Calabi flow
Let be a compact Kähler manifold and let be an initial Kähler metric. The Calabi flow is the evolution of Kähler metrics starting at by the negative gradient flow of the Calabi energy.
Chen's conjecture. The Calabi flow exists for all time.
This is one of the principal conjectures on the long-term behavior of the Calabi flow; the source does not state a resolution.
References
Primary source
Hongnian Huang, “Calabi flow on projective bundles, I”, arXiv:1511.06290 (2015).
Additional references
2 papers in this index state this conjecture (2012–2015). The statement above is taken from the most recent of them; the others are arXiv:1207.5964.
Progress summary
Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.
Solutions 1
RemarkAI-assistedClaimed by OpenAI. Claims a smooth Kahler metric in the Fubini–Study class of CP^10 whose Calabi flow develops unbounded scalar curvature in finite time, contradicting smooth long-time existence even in a class with a constant-scalar-curvature metric.See full solution
Claimed by OpenAI. Claims a smooth Kahler metric in the Fubini–Study class of CP^10 whose Calabi flow develops unbounded scalar curvature in finite time, contradicting smooth long-time existence even in a class with a constant-scalar-curvature metric.
Scope relative to this problem: The source reports finite-time scalar-curvature blowup for a smooth initial Kahler metric on CP^10 in the Fubini-Study class. This is the reported counterexample to unrestricted smooth all-time Calabi-flow existence even though the class contains a constant-scalar-curvature metric; it does not deny weak or generalized continuation.
GitHub repository: https://github.com/openai/math
- OpenAI-352-01-A-finite-time-singularity-of-Calabi-flow-on-projective-space.pdfOpen