PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 122, Number 3, November 1994 RIEMANNIAN METRICS WITH LARGE Xx B. COLBOISAND J. DODZIUK (Communicated by Peter Li) Abstract. We show that every compact smooth manifold of three or more dimensions carries a Riemannian metric of volume one and arbitrarily large first eigenvalue of the Laplacian. Let (Mn, g) be a compact, connected Riemannian manifold of n dimen- sions. The Laplacian Ag acting on functions on M has discrete spectrum. Let Xx(g) denote the smallest positive eigenvalue of A^ . Hersch [5] proved that Xx(g)vol(S2,g)<$n for every Riemannian metric g on the 2-sphere S2. In connection with this result, Berger [2] asked whether there exists a constant k(M) suchthat (1) Xx(g)vol(Mn,g)2l"<k(M) for any Riemannian metric on M. Yang and Yau [8] proved that the inequality above holds for a compact surface S of genus y with k(S) = &n(y + 1). Subsequently, numerous examples of manifolds were constructed for which (1) is false (cf. [3] for a discussion and references). In particular, for every n > 3, the sphere Sn admits metrics of volume one with Xx arbitrarily large [3, 6]. Bleecker conjectured in [3] that such metrics exist on every manifold M" if « > 3. In this note we give a very simple proof of Bleecker's conjecture using known examples and quite general principles. The same result has been proved independently by Xu [7] by a construction similar to ours. His argument, however, is much harder than our proof. Theorem 1. Every compact manifold M" with n > 3 admits metrics g of volume one with arbitrarily large Xx(g). Proof. The idea of the proof is very simple. We take a metric go on S" with vo^S", go) = 1 and Xx(go) > k + 1, where k is a large constant. We excise from S" a very small ball B(p, n) = Bn and form the connected sum of S" with M. The resulting manifold is diffeomorphic to M and has a submanifold £2, with smooth boundary, naturally identified with S" \ Bn . Let gx be an Received by the editors February 10, 1993. 1991 Mathematics Subject Classification. Primary 58G25; Secondary 53C21. This work was done while the second author enjoyed the hospitality of Forschungsinstitut für Mathematik at ETH Zürich. © 1994 American Mathematical Society 0002-9939/94 $1.00+ $.25 per page 905