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 LaplaceBeltrami 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.
Contents
Definition
Let M be an ndimensional closed Riemannian manifold. Let V(A) denote the volume of an ndimensional submanifold A and S(E) denote the n−1dimensional volume of a submanifold E (commonly called "area" in this context). The Cheeger isoperimetric constant of M is defined to be
where the infimum is taken over all smooth n−1dimensional 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 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 twodimensional 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 in terms of the isoperimetric constant h(M). Let M be an ndimensional closed Riemannian manifold whose Ricci curvature is bounded below by −(n−1)a^{2}, where a ≥ 0. Then
See also
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
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