﻿

# Cheeger constant

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

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

## 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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Cheeger constant (graph theory) — In mathematics, the Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a bottleneck . The Cheeger constant as a measure of bottleneckedness is of great interest in many… …   Wikipedia

• Cheeger bound — In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite state, discrete time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander… …   Wikipedia

• Jeff Cheeger — Infobox Scientist name = Jeff Cheeger caption = birth date = birth date|1943|12|1|df=y birth place = Brooklyn, U.S. death date = death place = residence = U.S. nationality = American field = Mathematician work institution = New York University… …   Wikipedia

• List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

• Goulot d'étranglement (informatique) —  Pour l’article homonyme, voir Goulot d étranglement (production).  Un goulot d étranglement est un point d un système limitant les performances globales, et pouvant avoir un effet sur les temps de traitement et de réponse. Les goulots… …   Wikipédia en Français

• Isoperimetric dimension — In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension …   Wikipedia

• Connectivity (graph theory) — In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) which need to be removed to disconnect the remaining nodes from each other[1]. It is… …   Wikipedia

• Conductance (graph) — For other uses, see Conductance. In graph theory the conductance of a graph G=(V,E) measures how well knit the graph is: it controls how fast a random walk on G converges to a uniform distribution. The conductance of a graph is often called the… …   Wikipedia

• Sectional curvature — In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two dimensional plane σp in the tangent space at p. It is the Gaussian curvature of… …   Wikipedia

• Collapsing manifold — For the concept in homotopy, see collapse (topology). In Riemannian geometry, a collapsing or collapsed manifold is an n dimensional manifold M that admits a sequence of Riemannian metrics gn, such that as n goes to infinity the manifold is close …   Wikipedia