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
- [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \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.[1]
Buser's inequality
Peter Buser proved an upper bound for [math]\displaystyle{ \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]\displaystyle{ \lambda_1(M)\leq 2a(n-1)h(M) + 10h^2(M). }[/math]
See also
- Cheeger constant (graph theory)
- Isoperimetric problem
- Spectral gap
References
- ↑ Buser, Peter (1978). "Über eine Ungleichung von Cheeger" (in de). Math. Z. 158 (3): 245–252. doi:10.1007/BF01214795.
- Buser, Peter (1982). "A note on the isoperimetric constant". Ann. Sci. École Norm. Sup. (4) 15 (2): 213–230. http://www.numdam.org/item?id=ASENS_1982_4_15_2_213_0.
- Cheeger, Jeff (1970). "A lower bound for the smallest eigenvalue of the Laplacian". in Gunning, Robert C.. Problems in analysis (Papers dedicated to Salomon Bochner, 1969). Princeton, N. J.: Princeton Univ. Press. pp. 195–199.
- 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.
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