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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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There is no such upper bound for the first eigenvalue, a flat torus is a counterexample. –  Anton Petrunin Jan 21 '13 at 6:20
    
Thank you!I should add "M is non-compact" –  jiangsaiyin Jan 21 '13 at 6:57
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1 Answer 1

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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