Families of non-tiling domains satisfying Pólya’s conjecture

We show the existence of classes of non-tiling domains satisfying Pólya’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 Pólya’s conjecture eventually, that is, for a sufficiently large order of the eigenvalues, and may be partioned into p non-overlapping isometric sub-domains, with p arbitrarily large, then there exists an order p₀ such that for p larger than p₀ all such sub-domains satisfy Pólya’s conjecture. In particular, this allows us to show that families of sectors of domains of revolution with analytic boundary, and thin cylinders satisfy Pólya’s conjecture, for instance. We also improve upon the Li–Yau constant for general cylinders in the Dirichlet case.