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arXiv · 2209.12671

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

Neal Coleman
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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.

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Neal Coleman. 2022-09-19. P\'olya's eigenvalue conjecture is false for spheres. https://arxiv.org/abs/2209.12671

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