Colling–Dunajski conjecture for all negative values of m

Prove that every closed m-quasi-Einstein manifold with constant parameter \(\lambda\leq0\) is trivial, meaning that its potential vector field \(X\) vanishes identically, for every \(m<0\).

Background

The paper discusses a conjecture of Colling and Dunajski concerning closed quasi-Einstein manifolds with constant λ≤0\lambda\leq0. Previously established results prove triviality only in the range m≤2−nm\leq 2-n. The conjecture proposes that the same rigidity conclusion should hold across the entire negative range m<0m<0.

The paper resolves the conjecture at the specific values m=−2m=-2 and m=−4m=-4, while leaving the general negative range unresolved. Its results therefore provide partial evidence rather than a complete proof of the conjecture.

References

Finally, in Section~\ref{sec:Witten}, we turn to a recent conjecture of Colling and Dunajski (see p.197), who expect Theorem~1.3 to remain valid for every $m<0$. More precisely, while their result establishes triviality for closed quasi-Einstein manifolds with $\lambda\leq0$ under the restriction $m\leq 2-n$, they conjecture that the same conclusion should hold throughout the entire negative range.

— Integral Identities and Rigidity of Generalized $m$-Quasi-Einstein Manifolds  (2609.11820 - Carvalho et al., 10 Sep 2026) in Section 1, Introduction