Sums of Laplace eigenvalues --- rotations and tight frames in higher dimensions

TL;DR

This paper proves that certain geometric shapes maximize sums of Laplace eigenvalues under volume and inertia constraints, using tight frames and rotation group symmetries, with conjectures extending to convex domains.

Contribution

It establishes extremal properties of simplexes, parallelepipeds, ellipsoids, and tori for Laplace eigenvalue sums, introducing a novel proof method involving tight frames and symmetry groups.

Findings

Regular simplex maximizes Laplace eigenvalue sums among simplexes.

Hypercube maximizes among parallelepipeds.

Ball maximizes among ellipsoids and tori.

Abstract

The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among ellipsoids for the ball, provided the volume and moment of inertia of an "inverse" body are suitably normalized. This result holds for Dirichlet, Robin and Neumann eigenvalues. Additionally, the cubical torus is shown to be maximal among flat tori. The method of proof involves tight frames for euclidean space generated by the orbits of the rotation group of the extremal domain. The ball is conjectured to maximize sums of Neumann eigenvalues among general convex domains, provided the volume and moment of inertia of the polar dual of the domain are suitably normalized.