Английская Википедия:Cheeger constant

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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

<math> h(M)=\inf_E \frac{S(E)}{\min(V(A), V(B))}, </math>

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 <math>\scriptstyle{\lambda_1(M)},</math> the smallest positive eigenvalue of the Laplacian on M, are related by the following fundamental inequality proved by Jeff Cheeger:

<math> \lambda_1(M)\geq \frac{h^2(M)}{4}. </math>

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.Шаблон:R

Buser's inequality

Peter Buser proved an upper bound for <math>\scriptstyle{\lambda_1(M)}</math> 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

<math> \lambda_1(M)\leq 2a(n-1)h(M) + 10h^2(M).</math>

See also

• Cheeger constant (graph theory)
• Isoperimetric problem
• Spectral gap

References

• Шаблон:Cite journal
• Шаблон:Cite book

• Шаблон:Cite book