A note on the isoperimetric constant

Abstract

Let M be a closed n-dimensional Riemannian manifold, h(M) its isoperimetric constant and λ1(M) the smallest positive eigenvalue of the Laplace-Beltrami operator Δ=−div grad. Cheeger's inequality is the bound λ1(M)≥14h2(M). The author's main result is an analogous upper bound. Specifically, for the Ricci curvature of M bounded below by −(n−1)δ2 (δ≥0), then indeed we have λ1≤2δ(n−1)h+10h2. The author cites three examples to show the limitations for such a result. First, for the general manifold (with no hypothesis on the curvature) a family of metrics is described with h→0 and λ1 bounded away from zero; second, for flat tori families exist with h→∞ and λ1∼h2; and finally, for compact Riemann surfaces with a metric of constant curvature −1, λ1>cgh, where cg depends only on the genus. A version of the main result is also presented for noncompact manifolds. As a further application of his techniques the author also considers the higher eigenvalues. His bounds have the same growth rate as Weyl's asymptotic law. Two independent proofs are given for the main result. The first relies on a regularity property of a current minimizing the isoperimetric ratio and a comparison theorem of Heintze and Karcher. The precise constants are obtained with this approach. The second proof, although more involved, is elementary in nature. An important ingredient is a lower bound for the isoperimetric constant of a starlike region in a Riemannian manifold. This last estimate is the basis for his consideration of the higher eigenvalues, and the noncompact case. Let M be a complete, noncompact Riemannian manifold with Ricci curvature bounded below by −(n−1)δ2; then λ1≤cδh, where c is a constant depending only on the dimension. The author notes that in the noncompact case h(M)≤δ(n−1), and hence the term h2 can be absorbed.