  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 $S^2$.
