Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...

, 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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

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

area Area is the quantity that expresses the extent of a region on the plane or on a curved 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 su ...

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 Eucl ...

that divides ''M'' into two disjoint pieces. In 1970, Jeff Cheeger proved an inequality that related the first nontrivial

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

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 ...

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 Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...

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; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...

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

\scriptstyle,

the smallest positive eigenvalue of the Laplacian on ''M'', are related by the following fundamental inequality proved by Jeff Cheeger: :

\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, 1978).

Buser's inequality

Peter Buser proved an upper bound for

\scriptstyle

in terms of the 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 measur ...

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

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

References

* * * * {{cite book , first=Alexander , last=Lubotzky , author-link=Alexander Lubotzky , title=Discrete groups, expanding graphs and invariant measures , series=Modern Birkhäuser Classics , mr=2569682 , publisher=Birkhäuser Verlag , location=Basel , year=1994 , isbn=978-3-0346-0331-7 , others=With an appendix by Jonathan D. Rogawski , doi=10.1007/978-3-0346-0332-4 Riemannian geometry

Definition

Cheeger's inequality

Buser's inequality

See also

References