Cheeger constant

This article is about 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. In 1971, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace–Beltrami operator on M to h(M). In 1982, Peter Buser proved a reverse version of this inequality, and the two inequalities put together are sometimes called the Cheeger-Buser inequality. These inequalities were highly influential not only in Riemannian geometry and global analysis, but also in the theory of Markov chains and in graph theory, where they have inspired the analogous Cheeger constant of a graph and the notion of conductance.

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

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

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

Cheeger's inequality

Jeff Cheeger proved^[1] a lower bound for the smallest positive eigenvalue ${\displaystyle {\lambda _{1}(M)}}$ of the Laplacian on M in term of what is now called the Cheeger isoperimetric constant h(M):

${\displaystyle \lambda _{1}(M)\geq {\frac {h^{2}(M)}{4}}.}$

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.^[2]

Buser's inequality

Peter Buser proved^[3] an upper bound for the smallest positive eigenvalue ${\displaystyle \lambda _{1}(M)}$ of the Laplacian on M in terms of the Cheeger 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

${\displaystyle \lambda _{1}(M)\leq 2a(n-1)h(M)+10h^{2}(M).}$

See also

• Cheeger constant (graph theory)
• Isoperimetric problem
• Spectral gap

Notes

1. Cheeger 1971.
2. Buser 1978.
3. Buser 1982.

References

• Buser, Peter (1978). "Über eine Ungleichung von Cheeger". Mathematische Zeitschrift (in German). 158 (3): 245–252. doi:10.1007/BF01214795. ISSN 0025-5874.

• Buser, Peter (1982). "A note on the isoperimetric constant". Annales scientifiques de l'École normale supérieure. 15 (2): 213–230. doi:10.24033/asens.1426. ISSN 0012-9593.

• Cheeger, Jeff (1971). "A Lower Bound for the Smallest Eigenvalue of the Laplacian". Problems in Analysis: A Symposium in Honor of Salomon Bochner (PMS-31). Princeton University Press. pp. 195–200. doi:10.1515/9781400869312-013. ISBN 978-1-4008-6931-2.