I'm looking for an upper bound on the first non-zero eigenvalue of the Laplace-Beltrami 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.