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

Abstract

We prove that every closed $m$-quasi-Einstein manifold $(M^n,g,X,λ)$ with constant $λ\leq0$ is trivial whenever $m\leq-2$. This establishes the Colling--Dunajski conjecture in the range $m\leq-2$ and, for $n\geq5$, extends the previously known range $m\leq2-n$. This result is placed within a broader study of closed generalized $m$-quasi-Einstein manifolds $(M^n,g,X,λ)$. 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 $\langle X,\nablaλ\rangle$. An identity for the drift Laplacian gives a new proof of the previously known triviality result for $m\leq-n$ and extends it to generalized $m$-quasi-Einstein manifolds under a pointwise sign condition on $\langle X,\nablaλ\rangle$. 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.