On the Invariant Spectrum of S−invariant Metrics on S

A theorem of J. Hersch (1970) states that for any smooth metric on S2, with total area equal to 4π, the first nonzero eigenvalue of the Laplace operator acting on functions is less than or equal to 2 (this being the value for the standard round metric). For metrics invariant under the standard S1-action on S2, one can restrict the Laplace operator to the subspace of S1-invariant functions and consider its spectrum there. The corresponding eigenvalues will be called invariant eigenvalues, and the purpose of this paper is to analyse their possible values. We first show that there is no general analogue of Hersch’s theorem, by exhibiting explicit families of S1-invariant metrics with total area 4π where the first invariant eigenvalue ranges through any value between 0 and ∞. We then restrict ourselves to S1-invariant metrics that can be embedded in R as surfaces of revolution. For this subclass we are able to provide optimal upper bounds for all invariant eigenvalues. As a consequence, we obtain an analogue of Hersch’s theorem with an optimal upper bound (greater than 2 and geometrically interesting). This subclass of metrics on S2 includes all S1-invariant metrics with non-negative Gauss curvature. One of the key ideas in the proofs of these results comes from symplectic geometry, and amounts to the use of the moment map of the S1-action as a coordinate function on S2.