In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h(M) defined in terms of the minimal area of a hypersurface that divides M into two disjoint pieces of equal volume. In 1970, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace-Beltrami operator on to . This proved to be a very influential idea in Riemannian geometry and global analysis and inspired an analogous theory for graphs.
The Cheeger constant. Let be an -dimensional closed Riemannian manifold. Let denote the volume of an -dimensional submanifold and denotes the -dimensional volume of an submanifold (commonly called “area” in this context).
Definition. The Cheeger isoperimetric constant of is defined as
where the infimum is taken over all smooth -dimensional submanifolds of which divide it into two disjoint submanifolds with boundary and . Isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume.
It is well-known that the Cheeger constant can be characterized by the following
The Cheeger inequality. The Cheeger constant and , the smallest positive eigenvalue of the Laplacian on , are related by the following fundamental inequality proved by Jeff Cheeger
This inequality is optimal in the following sense: for any , natural number and , there exists a two-dimensional Riemannian manifold with the isoperimetric constant and such that the th eigenvalue of the Laplacian is within from the Cheeger bound (Buser, 1978).
Following is the proof of the Cheeger inequality. I found this proof from a book due to Alexander Grigoryan (problem 10.8).
Proof. Replacing by we obtain
Taking in , we obtain the desired inequality.
The Buser inequality. Peter Buser proved an upper bound for in terms of the isoperimetric constant . Let be an -dimensional closed Riemannian manifold whose Ricci curvature is bounded below by , where . Then
For a proof of the Buser inequality, we refer the reader to a paper due to Ledoux published in Proc. Amer. Math. Soc. [here].