On the invariant spectrum of $S^1$-invariant metrics on $S^2$

A theorem of J. Hersch (1970) states that for any smooth metric on $S^2$, with total area equal to $4\pi$, 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 $S^1$-action on $S^2$, one can restrict the Laplace operator to the subspace of $S^1$-invariant functions and consider its spectrum there. The corresponding eigenvalues will be called invariant eigenvalues, and the purpose of this paper is to analyse its possible values. We first show that there is no general analogue of Hersch's theorem, by exhibiting explicit families of $S^1$-invariant metrics with total area $4\pi$ where the first invariant eigenvalue ranges through any value between 0 and $\infty$. We then restrict ourselves to $S^1$-invariant metrics that can be embedded in ${\bf R}^3$ 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 $S^2$ includes all $S^1$-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 $S^1$-action as a coordinate function on $S^2$.