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If M is a non-compact Riemannian manifold with Ric>=-(n-1),we know that the first eigenvalue λ(M)<=(n-1)^2/4.What if λ(M)=(n-1)^2/4,then M would be?Are there any paper on this topic?

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  • $\begingroup$ There is no such upper bound for the first eigenvalue, a flat torus is a counterexample. $\endgroup$ Jan 21, 2013 at 6:20

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Peter Li and Jiaping Wang have a series of papers on the structure of complete manifolds with
positive spectrum .Peter Li also has a survey article on this topic .You can find it on his home page .

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