Extremal $G$-invariant eigenvalues of the Laplacian of $G$-invariant metrics
| dc.creator | Colbois, Bruno | |
| dc.creator | Dryden, Emily B. | |
| dc.creator | Soufi, Ahmad El | |
| dc.date | 2007-02-19 | |
| dc.date.accessioned | 2026-07-07T08:47:47Z | |
| dc.date.available | 2026-07-07T08:47:47Z | |
| dc.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 $λ_k^G$. | |
| dc.description | To appear in Mathematische Zeitschrift | |
| dc.identifier | https://arxiv.org/abs/math/0702547 | |
| dc.identifier | http://arxiv.org/abs/math/0702547 | |
| dc.identifier | Mathematische Zeitschrift 258 (2007) 29 -- 41 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/143723 | |
| dc.subject | Differential Geometry | |
| dc.subject | Spectral Theory | |
| dc.subject | 58J50, 58E11, 35P15 | |
| dc.title | Extremal $G$-invariant eigenvalues of the Laplacian of $G$-invariant metrics | |
| dc.type | text |