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| | {{about|the Cheeger isoperimetric constant and Cheeger's inequality in Riemannian geometry|a different use|Cheeger constant (graph theory)}} |
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| | In [[Riemannian geometry]], the '''Cheeger isoperimetric constant''' of a [[compact space|compact]] [[Riemannian manifold]] ''M'' is a positive real number ''h''(''M'') defined in terms of the minimal [[surface area|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 [[graph (mathematics)|graph]]s. |
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| | == Definition == |
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| | Let ''M'' be an ''n''-dimensional [[closed manifold|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 |
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| | : <math> h(M)=\inf_E \frac{S(E)}{\min(V(A), V(B))}, </math> |
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| | 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''. Isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume. |
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| | == Cheeger's inequality == |
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| | 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: |
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| | : <math> \lambda_1(M)\geq \frac{h^2(M)}{4}. </math> |
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| | 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 ''k''th eigenvalue of the Laplacian is within ''ε'' from the Cheeger bound (Buser, 1978). |
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| | == Buser's inequality == |
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| | 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)''a''<sup>2</sup>, where ''a'' ≥ 0. Then |
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| | : <math> \lambda_1(M)\leq 2a(n-1)h(M) + 10h^2(M).</math> |
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| | == See also == |
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| | * [[Cheeger constant (graph theory)]] |
| | * [[Isoperimetric problem]] |
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| | == References == |
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| | * Peter Buser, ''A note on the isoperimetric constant''. Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213—230.{{MR|0683635}} |
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| | * Peter Buser, "Über eine Ungleichung von Cheeger". Math. Z. 158 (1978), no. 3, 245–252. {{MR|0478248}} |
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| | * [[Jeff Cheeger]], ''A lower bound for the smallest eigenvalue of the Laplacian''. Problems in analysis (Papers dedicated to [[Salomon Bochner]], 1969), pp. 195–199. Princeton Univ. Press, Princeton, N. J., 1970 {{MR|0402831}} |
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| | <!-- This is not ideal, since mostly devoted to graphs --> |
| | * [[Alexander Lubotzky]], ''Discrete groups, expanding graphs and invariant measures''. Progress in Mathematics, vol 125, Birkhäuser Verlag, Basel, 1994 |
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| | [[Category:Riemannian geometry]] |
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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 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.
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
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. Isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume.
Cheeger's inequality
The Cheeger constant h(M) and 
the smallest positive eigenvalue of the Laplacian on M, are related by the following fundamental inequality proved by Jeff Cheeger:
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 (Buser, 1978).
Buser's inequality
Peter Buser proved an upper bound for 
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
See also
References
- Peter Buser, A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213—230.Template:MR
- Peter Buser, "Über eine Ungleichung von Cheeger". Math. Z. 158 (1978), no. 3, 245–252. Template:MR
- Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian. Problems in analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton Univ. Press, Princeton, N. J., 1970 Template:MR
- Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures. Progress in Mathematics, vol 125, Birkhäuser Verlag, Basel, 1994