Global analysis, analysis on manifolds Spectral geometry Theorem

Cheeger's inequality for the first eigenvalue

Builds on Riemannian manifold Laplace–Beltrami operator Spectrum of the Laplace–Beltrami operator

Statement

Let $M$ be a compact connected Riemannian manifold without boundary, of dimension $n \ge 2$ . Consider the compact smooth hypersurfaces $\Sigma$ that divide $M$ into two disjoint open sets $A$ and $B$ with $M = A \cup \Sigma \cup B$ . Let $\operatorname{area}(\Sigma)$ be the $(n-1)$ -dimensional volume of $\Sigma$ . The Cheeger constant of $M$ is $h(M) = \inf_{\Sigma} \frac{\operatorname{area}(\Sigma)}{\min(\operatorname{vol}(A), \operatorname{vol}(B))}.$ Let $\lambda_1$ be the smallest nonzero eigenvalue of the Laplace–Beltrami operator of $M$ . Then $\lambda_1 \ge \frac{h(M)^2}{4}.$

Remarks

The proof starts from the variational characterisation of $\lambda_1$ as the infimum of $\int_M \abs{\nabla f}^2 / \int_M f^2$ over nonzero smooth functions $f$ with $\int_M f = 0$ . One applies the coarea formula to $g^2$ , where $g$ is the positive part of an eigenfunction shifted so that its support has at most half the volume. The bound from $h(M)$ on each level set, together with the Cauchy–Schwarz inequality, gives the factor $1/4$ . The inequality shows that a small $\lambda_1$ forces a small Cheeger constant, that is, a narrow bottleneck. A dumbbell, two round spheres joined by a thin tube, has a small Cheeger constant and also a small $\lambda_1$ . The reverse bound needs more hypotheses: Buser proved an upper bound for $\lambda_1$ in terms of $h(M)$ under a lower bound on the Ricci curvature. The Cheeger constant of a graph satisfies a discrete analogue of the inequality.

Going deeper

For a student

The inequality gives a lower bound on $\lambda_1$ from purely geometric data, without assumptions on curvature. Its discrete version for graphs is a basic tool for expander graphs, where the spectral gap and the edge expansion control each other. It complements curvature bounds such as the Lichnerowicz–Obata theorem, which need a positive lower bound on the Ricci curvature.

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Laplace–Beltrami operator concept
Riemannian manifold concept
Spectrum of the Laplace–Beltrami operator concept
Cheeger's inequality for the first eigenvalue statement

Tag	`JRGADFNBV2`
URI	`m2://JRGADFNBV2`
Kind	statement, theorem
Writing	W0 stub
MSC	58J50, 35P15, 53C21
Status	This copy of the site has no build data. Run `m2 status JRGADFNBV2`.
About	Spectrum of the Laplace–Beltrami operator
Uses	Riemannian manifold, Laplace–Beltrami operator, Spectrum of the Laplace–Beltrami operator

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