Ngô Quốc Anh

December 15, 2010

The Cheeger isoperimetric constant

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 19:30

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.

The Cheeger constant. Let $M$ be an $n$-dimensional closed Riemannian manifold. Let $V(A)$ denote the volume of an $n$-dimensional submanifold $A$ and $S(E)$ denotes the $n-1$-dimensional volume of an submanifold $E$ (commonly called “area” in this context).

Definition. The Cheeger isoperimetric constant of $M$ is defined as

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

where the infimum is taken over all smooth $n-1$-dimensional submanifolds $E$ of $M$ which divide it into two disjoint submanifolds with boundary $A$ and $B$. Isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume.

It is well-known that the Cheeger constant $h(M)$ can be characterized by the following

$\displaystyle h(M): = \mathop {\inf }\limits_{f \in C_0^\infty (M)\backslash \{ 0\} } \frac{{\int_M {\left| {\nabla f} \right|d\mu } }}{{\int_M {\left| f \right|d\mu } }}$.

The Cheeger inequality. The Cheeger constant $h(M)$ and $\lambda_1(M)$, 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 $\varepsilon>0$, there exists a two-dimensional Riemannian manifold $M$ with the isoperimetric constant $h(M) = h$ and such that the $k$th eigenvalue of the Laplacian is within $\varepsilon$ from the Cheeger bound (Buser, 1978).

Following is the proof of the Cheeger inequality. I found this proof from a book due to Alexander Grigoryan (problem 10.8).

Proof. Replacing $f$ by $f^2$ we obtain

$\displaystyle h(M) \leqslant \frac{{\int_M {\left| {\nabla ({f^2})} \right|d\mu } }}{{\int_M {{f^2}d\mu } }} = 2\frac{{\int_M {\left| f \right|\left| {\nabla f} \right|d\mu } }}{{\int_M {{f^2}d\mu } }} \leqslant 2\frac{{{{\left\| f \right\|}_{{L^2}}}{{\left\| {\nabla f} \right\|}_{{L^2}}}}}{{\left\| f \right\|_{{L^2}}^2}} = 2\frac{{{{\left\| {\nabla f} \right\|}_{{L^2}}}}}{{{{\left\| f \right\|}_{{L^2}}}}}$.

Taking $\inf$ in $f$, we obtain the desired inequality.

The Buser inequality. Peter Buser proved an upper bound for $\lambda_1(M)$ in terms of the isoperimetric constant $h(M)$. Let $M$ be an $n$-dimensional closed Riemannian manifold whose Ricci curvature is bounded below by $-(n-1)a^2$, where $a \geqslant 0$. Then

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

For a proof of the Buser inequality, we refer the reader to a paper due to Ledoux published in Proc. Amer. Math. Soc. [here].

Source: WikiPedia