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 kth 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

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