Cheeger constant
Constant in Riemannian geometry
- 2 min read
- 3 chapters
Related on Lably
Through Surface area
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. The isoperimetric constant may be defined more generally for noncompact Riemannian manifolds of finite volume.
If the manifold is compact with boundary, the infimum is taken over the submanifolds E which do not intersect the boundary, and the volume of the denominator is that of the disjoint piece whose boundary is only E. In the non-compact case with infinite volume, the infimum is taken over compact submanifolds E and the denominator corresponds to the volume of the bounded piece.
Sign in to keep reading
2 more chapters are free with a Lably account.
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):
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's inequality
Peter Buser proved an upper bound for the smallest positive eigenvalue 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 is bounded below by −(n−1)a2, where a ≥ 0. Then