Integral Identities and Rigidity of Generalized m-Quasi-Einstein Manifolds
Abstract
We prove that every closed m-quasi-Einstein manifold (Mn,g,X,λ) with constant λ≤0 is trivial whenever m≤−2. This establishes the Colling--Dunajski conjecture in the range m≤−2 and, for n≥5, extends the previously known range m≤2−n. This result is placed within a broader study of closed generalized m-quasi-Einstein manifolds (Mn,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 ⟨ X,∇λ⟩. An identity for the drift Laplacian gives a new proof of the previously known triviality result for m≤−n and extends it to generalized m-quasi-Einstein manifolds under a pointwise sign condition on ⟨ X,∇λ⟩. 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.
Related
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BibTeX
@misc{carvalho2026integral,
title = {{Integral Identities and Rigidity of Generalized $m$-Quasi-Einstein Manifolds}},
author = {Alcides de Carvalho and W. O. Costa-Filho},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11820},
eprint = {2609.11820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11820},
}
RIS
TY - UNPB
TI - Integral Identities and Rigidity of Generalized m-Quasi-Einstein Manifolds
AU - Alcides de Carvalho
AU - W. O. Costa-Filho
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.11820
UR - https://arxiv.org/abs/2609.11820
AB - We prove that every closed m-quasi-Einstein manifold (M^n,g,X,λ) with constant λ≤0 is trivial whenever m≤−2. This establishes the Colling--Dunajski conjecture in the range m≤−2 and, for n≥5, extends the previously known range m≤2−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 ⟨ X,∇λ⟩. An identity for the drift Laplacian gives a new proof of the previously known triviality result for m≤−n and extends it to generalized m-quasi-Einstein manifolds under a pointwise sign condition on ⟨ X,∇λ⟩. 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.
ER -