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

Abstract : 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 $\lambda_k^G$ 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 $\lambda_k^G$ admits no extremal metric under volume-preserving $G$-invariant deformations. If, moreover, $M$ has dimension at least three, then the functional $\lambda_k^G$ 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 $\lambda_k^G$.
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Mathematische Zeitschrift, Springer, 2007, 258, pp.29 -- 41
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Bruno Colbois, Emily Dryden, Ahmad El Soufi. Extremal $G$-invariant eigenvalues of the Laplacian of $G$-invariant metrics. Mathematische Zeitschrift, Springer, 2007, 258, pp.29 -- 41. 〈hal-00131806〉

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