Spectral Geometry of the Laplacian and Isoperimetic Constants: Cheeger's Inequality

Abstract:

In 1970, Cheeger bounded the lowest nonzero eigenvalue  λ1λ1 of the Laplacian on a Riemannain manifold using a geometrical property now known as the Cheeger isoperiemtric constant. Cheeger's inequality proved to be quite influential leading to further study as well as generalisations. In this talk, we review Cheeger's inequality and its background, and introduce the idea of higher Cheeger constants. We show that, under certain conditions, a Cheeger-like inequality holds for all eigenvalues λkλk using these constants.