I'm looking for an upper bound on the first nonzero eigenvalue of the LaplaceBeltrami operator on compact manifolds of dimension greater than four that have constant negative curvature. In particular, I would like to know whether or not there is an upper bound that is less than $\frac{1}{2}n^2  2$, where $n$ is the dimension, for the case when the sectional curvature is $1$. I can find quite a few papers on lower bounds, but the few I could find on upper bounds required information that is hard to obtain. Any help would be greatly appreciated.
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$\begingroup$ Do you want an upper bound that depends on the dimension only? And is $\frac{1}{2}n^22$ known to be an upper bound? $\endgroup$– Deane YangCommented Dec 16, 2012 at 1:20

3$\begingroup$ I don't know an explicit upper bound, but one can be obtained by combining the Margulis constant, which gives a lower bound on the maximal injectivity radius, and use an estimate of the Dirichlet eigenvalues on the ball of that radius to get an upper bound of the eigenvalue on the manifold. You might have a look at the book by GallotHulinLafontaine which gives an explicit estimate of the eigenvalue in terms of the injectivity radius (I don't have a copy now to give the chapter). $\endgroup$– Ian AgolCommented Dec 16, 2012 at 6:09

3$\begingroup$ What about Buser inequality? (this is a "converse inequality" to the Cheeger inequality). $\endgroup$– AsafCommented Dec 16, 2012 at 7:25
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Did you look in Isaac Chavel's book 'Eigenvalues in Riemannian geometry'? It contains various bounds for eigenvalues although at a quick glance I didn't see anything for your particular setting.