On the Unboundedness of the First Eigenvalue of the Laplacian for G-Invariant Metrics

Abstract: In this note we partially answer a question posed by Colbois, Dryden, and El Soufi. Consider the space of constant-volume Riemannian metrics on a connected manifold M which are invariant under the action of a discrete Lie group G. We show that the first eigenvalue of the Laplacian is not bounded above on this space, provided M = S^n, G acts freely, and S^n/G with the round metric admits a Killing vector field of constant length, or provided M is a compact Lie group not equal to T^n, and G is a discrete subgroup of M.