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. 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.

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

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

The Cheeger constant h(M) and ${\displaystyle \scriptstyle {\lambda _{1}(M)},}$ the smallest positive eigenvalue of the Laplacian on M, are related by the following fundamental inequality proved by Jeff Cheeger:

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

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 (Buser, 1978).

Buser's inequality

Peter Buser proved an upper bound for ${\displaystyle \scriptstyle {\lambda _{1}(M)}}$ 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

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

See also

• Cheeger constant (graph theory)
• Isoperimetric problem

References

• Buser, Peter (1982). "A note on the isoperimetric constant". Ann. Sci. École Norm. Sup. (4). 15 (2): 213–230. MR 0683635.
• Buser, Peter (1978). "Über eine Ungleichung von Cheeger" [On an inequality of Cheeger]. Math. Z. (in German). 158 (3): 245–252. doi:10.1007/BF01214795. MR 0478248.
• Cheeger, Jeff (1970). "A lower bound for the smallest eigenvalue of the Laplacian". In Gunning, Robert C. (ed.). Problems in analysis (Papers dedicated to Salomon Bochner, 1969). Princeton, N. J.: Princeton Univ. Press. pp. 195–199. MR 0402831.

• Lubotzky, Alexander (1994). Discrete groups, expanding graphs and invariant measures. Modern Birkhäuser Classics. With an appendix by Jonathan D. Rogawski. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0346-0332-4. ISBN 978-3-0346-0331-7. MR 2569682.