Sturmliouville estimates for the spectrum and cheeger constant

Abstract

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, λκ(M), in terms of the eigenvalues of a Sturm-Liouville problem which depends on h(M). We then compare the Weyl asymptotic of λκ(M) given by the works of Cheng, Gromov, and Berard-Besson-Gallot to the asymptotics of our Sturm-Liouville problems given by Atkinson-Mingarelli.