The study of extremal properties of the spectrum often involves restricting
the metrics under consideration. Motivated by the work of Abreu and Freitas in
the case of the sphere S2 endowed with S1-invariant metrics, we consider
the subsequence λkG of the spectrum of a Riemannian manifold M
which corresponds to metrics and functions invariant under the action of a
compact Lie group G. If G has dimension at least 1, we show that the
functional λkG admits no extremal metric under volume-preserving
G-invariant deformations. If, moreover, M has dimension at least three,
then the functional λkG is unbounded when restricted to any conformal
class of G-invariant metrics of fixed volume. As a special case of this, we
can consider the standard O(n)-action on Sn; however, if we also require the
metric to be induced by an embedding of Sn in Rn+1, we get an
optimal upper bound on λkG.Comment: To appear in Mathematische Zeitschrif