P\'olya's eigenvalue conjecture is false for spheres

TL;DR

The paper demonstrates that Pólya's eigenvalue conjecture does not hold for spheres by analyzing the Laplace spectrum and Weyl function, showing the conjecture's limitations in positively curved Riemannian manifolds.

Contribution

It provides a counterexample to Pólya's eigenvalue conjecture on spheres, revealing the conjecture's failure in certain curved geometries.

Findings

Pólya's conjecture does not hold for spheres.

Spectral comparison shows deviation from conjecture predictions.

Counterexamples in positively curved manifolds are identified.

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 P\'olya's eigenvalue conjecture holds in general for Riemannian manifolds with positive sectional curvature.