A simple analytic proof of an inequality by P. Buser

We present a simple analytic proof of the inequality of P. Buser showing the equivalence of the first eigenvalue of a compact Riemannian manifold without boundary and Cheeger's isoperimetric constant under a lower bound on the Ricci curvature. Our tools are the Li-Yau inequality and ideas of Varopoulos in his functional approach to isoperimetric inequalities and heat kernel estimates on groups and manifolds. The method is easily modified to yield a logarithmic isoperimetric inequality involving the hypercontractivity constant of the manifold. 1. BUSER'S INEQUALITY Throughout this paper, M will denote a compact Riemannian manifold without boundary of dimension n. We denote by ,u the normalised Riemannian measure on M, by A the Laplace operator, and by Vf the gradient of a smooth function f on M with Riemannian length lVfl. The first nontrivial eigenvalue Al of the Laplacian is characterised via the min-max theorem by the Poincare type inequality AlJ 2dl? 0, for all f with f f da = 0, where 11 . I[P is the LP-norm (1 0, is the heat semigroup on M. In 1970, Cheeger [C] introduced an isoperimetric constant to bound below the first eigenvalue Al . Set h = inf a((A) u (A) where the infimum runs over all open subsets A with ,u(A) < I and smooth boundary AA, and where a(.) denotes the (n 1)-dimensional measure. Received by the editors October 14, 1992. 1991 Mathematics Subject Classification. Primary 58G1 1, 58G25, 53C99, 49Q15. ( 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page