Bruno Colbois

Riemannian metrics with large

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 dimensions. The Laplacian Ag acting on functions on M has discrete spectrum. Let A1 (g) denote the smallest positive eigenvalue of Ag . Hersch [5] proved that AIl(g)vol(S2, g) 3, the sphere Sn admits metrics of volume one with A1 arbitrarily large [3, 6]. Bleecker conjectured in [3] that such metrics exist on every manifold Mn if n > 3. In this note we give a very simple proof of Bleeckers 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 Mn with n > 3 admits metrics g of volume one with arbitrarily large AI (g) . Proof. The idea of the proof is very simple. We take a metric go on Sn with vol(Sn, go) = 1 and AL (go) > k + 1, where k is a large constant. We excise from Sn a very small ball B(p, t) = B. and form the connected sum of Sn with M. The resulting manifold is diffeomorphic to M and has a submanifold Q, with smooth boundary, naturally identified with Sn \ Bi . Let g1 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 fur Mathematik at ETH Zurich. @ 1994 American Mathematical Society 0002-9939/94

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