Let M be an n-dimensional closed Riemannian manifold. Let V(A) denote the volume of an n-dimensional submanifold A and S(E) denote the n−1-dimensional volume of a submanifold E (commonly called "area" in this context). The Cheeger isoperimetric constant of M is defined to be

${\displaystyle h(M)=\inf _{E}{\frac {S(E)}{\min(V(A),V(B))}},}$ 

where the infimum is taken over all smooth n−1-dimensional submanifolds E of M which divide it into two disjoint submanifolds A and B. 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 ${\displaystyle {\lambda _{1}(M)}}$  of the Laplacian on M in term of what is now called the Cheeger isoperimetric constant h(M):

${\displaystyle \lambda _{1}(M)\geq {\frac {h^{2}(M)}{4}}.}$ 

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 ${\displaystyle \lambda _{1}(M)}$  of the Laplacian on M in terms of the Cheeger isoperimetric constant h(M). Let M be an n-dimensional closed Riemannian manifold whose Ricci curvature is bounded below by −(n−1)a², where a ≥ 0. Then

${\displaystyle \lambda _{1}(M)\leq 2a(n-1)h(M)+10h^{2}(M).}$ 

• Cheeger constant (graph theory)
• Isoperimetric problem
• Spectral gap