In 1971, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace–Beltrami operator on M to h(M).
The isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume.
Jeff Cheeger proved[1] a lower bound for the smallest positive eigenvalue
of the Laplacian on M in term of what is now called the Cheeger isoperimetric constant h(M): This inequality is optimal in the following sense: for any h > 0, natural number k, and ε > 0, there exists a two-dimensional Riemannian manifold M with the isoperimetric constant h(M) = h and such that the kth eigenvalue of the Laplacian is within ε from the Cheeger bound.
[2] Peter Buser proved[3] an upper bound for the smallest positive eigenvalue