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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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  • [1] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété Riemannienne, Lecture Notes in Math., vol. 194, Springer-Verlag, Berlin and New York, 1971. MR 0282313 (43:8025)
  • [2] S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d'une variété Riemannienne, J. Math. Pures Appl. 54 (1975), 259-284. MR 0454884 (56:13128)
  • [3] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. I, Interscience, New York, 1963. MR 0152974 (27:2945)
  • [4] A. Lichnerowicz, Géométrie des groupes de transformations, Dunod, Paris, 1958.
  • [5] D. Perrone, Varietà conformemente piatte e geometria spettrale, Riv. Mat. Univ. Parma (4) 8 (1982) (preprint in Quaderni Ist. Mat. Univ. Lecce Q.20, 1980). MR 706856 (84i:58123)
  • [6] U. Simon and H. Wissner, Geometry of the Laplace operator, Technische Universität Berlin, Preprint Reihe Mathematik, No. 73, 1980. MR 698272 (84f:58113)
  • [7] M. Tani, On a compact conformally flat space with positive Ricci curvature, Tôhoku Math. J. 19 (1967), 227-231. MR 0220213 (36:3279)

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Article copyright: © Copyright 1982 American Mathematical Society

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