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Buser's inequality gives an upper bound on the first nonzero eigenvalue of the Laplacian of a closed manifold   $M$   in terms of the Cheeger constant   $h(M)$  . Agol later gave a quantitative improvement of Buser's inequality. Agol's result is less transparent since it is given implicitly by a set of equations, one of which is a differential equation Agol could not solve except when   $M$   is 3D. We show that a substitution transforms Agol's differential equation into the Riemann differential equation. Then, we give a proof of Agol's result and also generalize it using the Sturm–Liouville theory. Under the same assumptions on   $M$  , we are able to give upper bounds on the higher eigenvalues of   $M$  ,   $\lambda _k(M)$  , in terms of the eigenvalues of a Sturm–Liouville problem which depends on   $h(M)$  . We then compare the Weyl asymptotic of   $\lambda _k(M)$   given by the works of Cheng, Gromov, and Bérard–Besson–Gallot to the asymptotics of our Sturm–Liouville problems given by Atkinson–Mingarelli.

Sturm–Liouville Estimates for the Spectrum and Cheeger Constant