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Colling–Dunajski conjecture

For every closed generalized mm-quasi-Einstein manifold (Mn,g,X,λ)(M^n,g,X,\lambda) with m∈R∖{0}m\in\mathbb{R}\setminus\{0\} and constant λ≤0\lambda\leq 0, satisfying Ric⁡g+12LXg−1mX♭⊗X♭=λg\operatorname{Ric}_g+\frac{1}{2}\mathcal{L}_Xg-\frac{1}{m}X^\flat\otimes X^\flat=\lambda g, the structure is trivial; equivalently, X≡0X\equiv 0 and Ric⁡g=λg\operatorname{Ric}_g=\lambda g.

References

Primary source

arXiv

Additional references

differential-geometrygeometric-analysis
Posted · edited

Progress summary

Refreshed
Claimed progress

A new unrefereed manuscript claims further progress on the rigidity conjecture, but the full conjecture remains open.

The Colling–Dunajski conjecture predicts rigidity for generalized quasi-Einstein structures. Colling and Dunajski’s 2025 work established substantial parameter ranges, without settling the entire conjecture.

Known results

  • Colling and Dunajski (2025): a commuting Killing field exists for closed manifolds when m≥2m\ge 2 or m≤2−nm\le 2-n.
  • For n=2n=2, the same conclusion holds when m≥2m\ge 2 or m<0m<0.
  • Their work gives a classification of compact quasi-Einstein 22-manifolds for m∉(0,2)m\notin(0,2).
  • Colling and Dunajski (2023) obtained additional rigidity results in the divergence-free case.

September 2026 claimed advance

Alcides de Carvalho and W. O. Costa-Filho report that integral and differential identities yield the conjectured rigidity at two exceptional parameter values. The manuscript is unrefereed and does not claim coverage of the full conjecture.

Current status (as of September 2026): substantial parameter ranges are established, and two further parameter values are claimed in an unrefereed manuscript, but the full Colling–Dunajski conjecture remains open.

Sources

Solutions 0

No solutions have been posted yet.