Families of non-tiling domains satisfying P\'olya's conjecture

TL;DR

This paper demonstrates the existence of non-tiling domains in any dimension that satisfy Pólya's conjecture, including sectors of revolution and thin cylinders, and improves the Li-Yau constant for cylinders.

Contribution

It introduces a general criterion for non-tiling domains to satisfy Pólya's conjecture and applies it to specific geometric families, extending previous results.

Findings

Families of sectors of domains of revolution satisfy Pólya's conjecture.

Thin cylinders satisfy Pólya's conjecture.

Li-Yau constant for general cylinders is improved.

Abstract

We show the existence of classes of non-tiling domains satisfying P\'{o}lya's conjecture in any dimension, in both the Euclidean and non-Euclidean cases. This is a consequence of a more general observation asserting that if a domain satisfies P\'{o}lya's conjecture eventually, that is, for a sufficiently large order of the eigenvalues, and may be partitioned into $p$ non-overlapping isometric sub-domains, with $p$ arbitrarily large, then there exists an order $p_{0}$ such that for $p$ larger than $p_{0}$ all such sub-domains satisfy P\'{o}lya's conjecture. In particular, this allows us to show that families of sectors of domains of revolution with analytic boundary, and thin cylinders satisfy P\'{o}lya's conjecture, for instance. We also improve upon the Li-Yau constant for general cylinders in the Dirichlet case.