2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/143723The 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 $S^2$ endowed with $S^1$-invariant metrics, we consider the subsequence $λ_k^G$ 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 $λ_k^G$ admits no extremal metric under volume-preserving $G$-invariant deformations. If, moreover, $M$ has dimension at least three, then the functional $λ_k^G$ 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 $S^n$; however, if we also require the metric to be induced by an embedding of $S^n$ in $\mathbb{R}^{n+1}$, we get an optimal upper bound on $λ_k^G$.To appear in Mathematische ZeitschriftDifferential GeometrySpectral Theory58J50, 58E11, 35P15text