Cheeger constant


In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h defined in terms of the minimal area of a hypersurface that divides M into two disjoint pieces. In 1970, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace–Beltrami operator on M to h. This proved to be a very influential idea in Riemannian geometry and global analysis and inspired an analogous theory for graphs.

Definition

Let M be an n-dimensional closed Riemannian manifold. Let V denote the volume of an n-dimensional submanifold A and S denote the n−1-dimensional volume of a submanifold E. The Cheeger isoperimetric constant of M is defined to be
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.

Cheeger's inequality

The Cheeger constant h and the smallest positive eigenvalue of the Laplacian on M, are related by the following fundamental inequality proved by Jeff Cheeger:
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 = h and such that the kth eigenvalue of the Laplacian is within ε from the Cheeger bound.

Buser's inequality

Peter Buser proved an upper bound for in terms of the isoperimetric constant h. Let M be an n-dimensional closed Riemannian manifold whose Ricci curvature is bounded below by −a2, where a ≥ 0. Then