Extremal $G$-invariant eigenvalues of the Laplacian of $G$-invariant metrics

Colbois, Bruno; Dryden, Emily B.; Soufi, Ahmad El

Extremal $G$-invariant eigenvalues of the Laplacian of $G$-invariant metrics

Authors

Colbois, Bruno

Dryden, Emily B.

Soufi, Ahmad El

Description

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 $S^2$ 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 Zeitschrift