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

TL;DR

This paper demonstrates that the first eigenvalue of the Laplacian can be arbitrarily large for G-invariant metrics on certain manifolds, under specific symmetry and geometric conditions.

Contribution

It provides partial answers to a question about eigenvalue bounds, showing unboundedness in new classes of G-invariant metrics on manifolds like spheres and compact Lie groups.

Findings

First eigenvalue is unbounded on the space of G-invariant metrics for spheres with free G-actions.

Unboundedness also holds for certain compact Lie groups with discrete subgroups G.

Conditions involve existence of Killing vector fields of constant length and specific geometric structures.

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.