Pólya's eigenvalue conjecture is false for spheres

Abstract

By comparing the Laplace spectrum of the sphere ${\mathbb{S}}^n$ to its Weyl function $w(x)=\frac{{\omega }_n}{(2\pi {)}^n}|{\mathbb{S}}^n|x^{n/2}$, we show that no analogue of Pólya's eigenvalue conjecture holds in general for Riemannian manifolds with positive sectional curvature.

5 pages, 1 figure