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A lower bound for the spectrum of the Laplacian in terms of sectional and Ricci curvature


Author: Alberto G. Setti
Journal: Proc. Amer. Math. Soc. 112 (1991), 277-282
MSC: Primary 58G25
DOI: https://doi.org/10.1090/S0002-9939-1991-1043421-3
MathSciNet review: 1043421
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Abstract: Let $ M$ be an $ n$-dimensional, complete, simply connected Riemannian manifold. In this paper we show that if the sectional curvature is bounded above by $ - k \leq 0$ and the Ricci curvature is bounded above by $ - \alpha \leq 0$, then the spectrum of the Laplacian on $ M$ is bounded below by $ [\alpha + (n - 1)(n - 2)k]/4$. This improves a previous result due to H. P. McKean.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1043421-3
Article copyright: © Copyright 1991 American Mathematical Society

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