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.

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.