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Type-I singularity conjecture for Kähler–Ricci flow on compact Kähler surfaces

For every compact Kähler surface MM and every maximal solution g(t)g(t) of the Kähler–Ricci flow ∂∂tg(t)=−Ric⁡(g(t))\frac{\partial}{\partial t}g(t)=-\operatorname{Ric}(g(t)) on [0,T)[0,T) with T<∞T<\infty, there exists a constant C<∞C<\infty such that sup⁡x∈M∣Rm⁡(g(t))∣g(t)(x)≤CT−t\sup_{x\in M}\lvert\operatorname{Rm}(g(t))\rvert_{g(t)}(x)\leq \frac{C}{T-t} for all 0≤t<T0\leq t<T.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Collapsed finite-time singularities are Type I

    Every collapsed finite-time singularity of the Kähler–Ricci flow on a compact Kähler surface satisfies the Type-I curvature bound sup⁡x∈M∣Rm⁡(g(t))∣g(t)(x)≤C/(T−t)\sup_{x\in M}\lvert\operatorname{Rm}(g(t))\rvert_{g(t)}(x)\leq C/(T-t) for some constant C<∞C<\infty.

    source: Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

References

Primary source

arXiv

Additional references

complex-differential-geometrygeometric-analysisricci-flow
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims that every finite-time singularity on a compact Kähler surface has the expected Type-I behavior, but this has not been independently verified.

The conjecture asserts that every finite-time singularity of Kähler–Ricci flow on a compact Kähler surface is Type I, which would complete the classification in complex dimension two.

Known results

  • Cifarelli, Conlon, and Deruelle (2022) classified important Type-I limits, including the Feldman–Ilmanen–Knopf soliton in the non-collapsed case, while identifying a remaining collapsing possibility.
  • The same 2022 work reduced the remaining classification to the existence of a new toric shrinking soliton.
  • The non-collapsed case was proved Type I by Cifarelli, Conlon, and Deruelle (2025), with the contracted (−1)(-1)-curve model identified as the Feldman–Ilmanen–Knopf soliton.

September 2026 claimed completion

Tongxin Xu and Zhenlei Zhang's preprint asserts that collapsed finite-time singularities are Type I, while another September 2026 arXiv preprint states the full compact-surface theorem and classifies the tangent flows. Together these claims would settle the conjecture, but they remain supported only by unrefereed preprints.

Current status (as of September 2026): The non-collapsed case and several Type-I model classifications are established, while the claimed Type-I proof for the collapsed case and hence the full conjecture remain unverified.

Sources

Solutions 0

No solutions have been posted yet.