Cheeger constant

This article discusses the Cheeger isoperimetric constant and Cheeger's inequality in Riemannian geometry. For a different use, see Cheeger constant (graph theory).

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 M to h(M). 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(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

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

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. Isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume.

Cheeger's inequality

The Cheeger constant h(M) and $\scriptstyle{\lambda_1(M)},$ the smallest positive eigenvalue of the Laplacian on M, are related by the following fundamental inequality proved by Jeff Cheeger:

$\lambda_1(M)\geq \frac{h^2(M)}{4}.$

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 (Buser, 1978).

Buser's inequality

Peter Buser proved an upper bound for $\scriptstyle{\lambda_1(M)}$ in terms of the 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

$\lambda_1(M)\leq 2a(n-1)h(M) + 10h^2(M).$

See also

• Isoperimetric problem

References

• Peter Buser, A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213—230.MR0683635

• Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian. Problems in analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton Univ. Press, Princeton, N. J., 1970 MR0402831

• Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures. Progress in Mathematics, vol 125, Birkhäuser Verlag, Basel, 1994