Integral Identities and Rigidity of Generalized $m$-Quasi-Einstein Manifolds

Abstract

We prove that every closed $m$-quasi-Einstein manifold $(M^n,g,X,\lambda )$ with constant $\lambda \le 0$ is trivial whenever $m\le -2$. This establishes the Colling--Dunajski conjecture in the range $m\le -2$ and, for $n\ge 5$, extends the previously known range $m\le 2-n$. This result is placed within a broader study of closed generalized $m$-quasi-Einstein manifolds $(M^n,g,X,\lambda )$. We establish differential and integral identities for such manifolds. These identities yield criteria for conformality, the Killing condition, and triviality, and provide a unified framework for several rigidity phenomena. After this, we recast the integrated Bochner formula as a Witten-type Hodge-energy identity. The cancellation of its quartic term at $m=-2$ leads to a triviality theorem in the generalized setting under a natural sign condition on the integral of $⟨X,\nabla \lambda ⟩$. An identity for the drift Laplacian gives a new proof of the previously known triviality result for $m\le -n$ and extends it to generalized $m$-quasi-Einstein manifolds under a pointwise sign condition on $⟨X,\nabla \lambda ⟩$. Finally, complementing our criterion characterizing when a conformal potential field is Killing, we exhibit in the appendix a closed generalized $m$-quasi-Einstein manifold whose potential field is conformal but non-Killing.