Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

, the Cheeger isoperimetric constant of a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

''M'' is a positive real number ''h''(''M'') defined in terms of the minimal

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

of a

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

that divides ''M'' into two disjoint pieces. In 1971, Jeff Cheeger proved an inequality that related the first nontrivial

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

of the

Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft ...

on ''M'' to ''h''(''M''). In 1982, Peter Buser proved a reverse version of this inequality, and the two inequalities put together are sometimes called the ''Cheeger-Buser inequality''. These inequalities were highly influential not only in Riemannian geometry and global analysis, but also in the theory of Markov chains and in

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, where they have inspired the analogous Cheeger constant of a graph and the notion of conductance.

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 :

h(M)=\inf_E \frac,

where the

infimum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...

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

Jeff Cheeger proved a lower bound for the smallest positive eigenvalue

of the Laplacian on ''M'' in term of what is now called the Cheeger isoperimetric constant ''h''(''M''): :

\lambda_1(M)\geq \frac.

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's inequality

Peter Buser proved an upper bound for the smallest positive eigenvalue

\lambda_1(M)

of the Laplacian on ''M'' in terms of the Cheeger isoperimetric constant ''h''(''M''). Let ''M'' be an ''n''-dimensional closed Riemannian manifold whose

Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...

is bounded below by −(''n''−1)''a''², where ''a'' ≥ 0. Then :

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

Notes

References

* * * {{refend Riemannian geometry

Definition

Cheeger's inequality

Buser's inequality

See also

Notes

References