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On the minimal eigenvalue of the Laplacian operator for $ p$-forms in conformally flat Riemannian manifolds


Author: Domenico Perrone
Journal: Proc. Amer. Math. Soc. 86 (1982), 103-108
MSC: Primary 53C20; Secondary 58G25
MathSciNet review: 663876
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Abstract: Let $ (M,g)$ be a compact orientable conformally flat Riemannian manifold and $ ^p{\lambda _1}$ the minimal eigenvalue of the Laplacian operator for $ p$-forms. We prove that if there exists a positive constant $ K$ such that $ \rho \geqslant Kg$, where $ \rho $ is the Ricci tensor of $ M$, then $ ^p{\lambda _1} \geqslant Kp(n - p + 1)/(n - 1)$ for each $ p$, $ 1 \leqslant p \leqslant n/2$, $ (n = \dim M)$; moreover if the equality holds for some $ p$ then $ M$ is of constant curvature $ \sigma = K/(n - 1)$.


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DOI: https://doi.org/10.1090/S0002-9939-1982-0663876-8
Article copyright: © Copyright 1982 American Mathematical Society