PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 7, Pages 2183–2192 S 0002-9939(05)07802-0 Article electronically published on February 15, 2005 BOUNDEDNESS OF THE FIRST EIGENVALUE OF THE p-LAPLACIAN ANA-MARIA MATEI (Communicated by Jozef Dodziuk) Abstract. We prove that for any p> 1, any compact manifold of three or more dimensions carries Riemannian metrics of volume one with the first eigenvalue of the p-Laplacian arbitrarily large. Introduction The p-Laplace operator has been extensively studied in recent years, especially in the context of a bounded domain in R n . Recently, there has been an increasing interest in the study of this operator – and in particular of its first eigenvalue – in the more general setting of Riemannian manifolds. Let M be a compact connected manifold. The p-Laplace operator (p> 1) associated to a Riemannian metric g on M is given by ∆ p f := δ(|df | p-2 df ) , where δ = -div g is the adjoint of d for the L 2 -norm induced by g on the space of differential forms. This operator can be viewed as a natural generalization of the well-known Laplace-Beltrami operator which corresponds to p = 2. When ∂M = ∅, the nonlinear partial differential equation (0.1) ∆ p f = λ|f | p-2 f characterizes the critical points of the p-energy functional E p (f )=  M |df | p ν g under the constraint  M |f | p ν g =1(ν g denotes the Riemannian volume element induced by g). The real numbers λ for which this equation has nontrivial solutions are called eigenvalues of ∆ p . They represent the critical energy levels. The associated solu- tions, i.e. the critical functions, are called eigenfunctions. Obviously, zero is an eigenvalue of ∆ p , the associated eigenfunctions being the constant functions. The set σ p (M,g) of the nonzero eigenvalues is a nonempty, unbounded subset of (0, ∞) [6]. Its infimum λ 1,p (M,g) = inf σ p (M,g) is itself a positive eigenvalue called the first eigenvalue of ∆ p and has the following variational characterization [14] : (0.2) λ 1,p (M,g) = inf   M |df | p ν g  M |f | p ν g | f ∈ W 1,p (M ) \{0},  M |f | p-2 fν g =0  . Received by the editors March 21, 2004 and, in revised form, April 8, 2004. 2000 Mathematics Subject Classification. Primary 58C40; Secondary 58J50. Key words and phrases. p-Laplacian, eigenvalue. c 2005 American Mathematical Society Reverts to public domain 28 years from publication 2183