On Cheeger's Inequality

In Ch], Cheeger proved the following general lower bound for the rst eigenvalue 1 of a closed Riemannian manifold: Theorem ((Ch]): 1 1 4 h 2 ; where h = inf N area(N) min(vol(A); vol(B)) where N runs over (possibly disconnected) hypersurfaces of M which divide M into two pieces A and B, and where area denotes (n ? 1)-dimensional volume, and vol denotes n-dimensional volume, where n = dim(M). h(M) is called the Cheeger constant of M. Cheeger's inequality is quite straightforward to prove, and is essentially the co-area formula of geometric measure theory. It is therefore surprising that the inequality plays such a crucial role in the study of the geometry of the Laplace operator, see Bu3]. Indeed, one has the following general upper bound for 1 in terms of h, due to Peter Buser Bu]: where c 1 ; c 2 depend only on a lower bound on the Ricci curvature of M. Thus, from a qualitative point of view, 1 and h are essentially the same thing, in the sense that one tends to zero if and only if the other does (in the presence of bounded curvature). We observe that Cheeger's inequality is true, and is proved in exactly the same way, when M is a complete, non-compact manifold, or a manifold with boundary and either Dirichlet or Neumann boundary conditions, provided one interprets 1 , and h correctly. It has therefore been an interesting question to understand, in a general way, how sharp Cheeger's inequality really is. A major problem in coming to terms with this question has been that, for the most part, Cheeger's inequality is the only generally useful method known for estimating 1 from below. In this paper, we will explore this question in three ways. First of all, by 1 a celebrated theorem of Selberg Se], there are general lower bounds 1 (S p) 3 16 for certain arithmetic Riemann surfaces S p , which we will discuss below. Selberg raised the question of whether 1 (S p) 1 4 for these surfaces, and it was suggested in Bi] that perhaps one could demonstrate this by showing that h(S p) 1 for these surfaces. We will show that this is not the case, and indeed h(S p) is so small for these surfaces that one cannot even obtain Selberg's 3 16 bound via Cheeger's constant: Theorem 1.1: For p 1(mod …