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 λ G k 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 G k admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional λ G k 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 R n+1 , we get an optimal upper bound on λ G k .

Maximal spectral surfaces of revolution converge to a catenoid

“…They bounded the S 1 -invariant eigenvalues of a surface of revolution in R 3 in terms of the eigenvalues of a disc. Colbois, Dryden, and El Soufi [16] extended this to higher dimensions. These results were motivated by Hersch [17], who showed that on a sphere, the round metric maximizes the first non-zero eigenvalue among metrics of given area.…”

On the invariant spectrum on

“…In , Colbois, Dryden and El Soufi considered a more general situation. Namely, the subsequence 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 dimension of G is at least 1, they showed that the functional admits no extremal metric under volume‐preserving G ‐invariant deformations, cf. [, Theorem ]. If, moreover, M has dimension at least 3, they proved that the functional is unbounded when restricted to any conformal class of G ‐invariant metrics of fixed volume, cf.…”

Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds

“…To study extremal properties of the spectrum, it is therefore necessary to impose additional constraints, either of an intrinsic or an extrinsic nature. For example, we can assume that the induced metric g preserves a conformal class of metrics [7,12,20], a symplectic or a Kähler structure [2,27], the action of a Lie group [1,6,14], etc. Regarding results with constraints of extrinsic type, a well-known example is given by Reilly's inequality [13,29]:…”