Improvement of Pólya's conjecture for balls and cylinders

TL;DR

The paper advances spectral geometry by sharpening Pólya-type bounds and Weyl remainder estimates for non-tiling domains — disks, balls, and cylinders. It extends analytic regimes for Neumann bounds on the disk, provides explicit improved Dirichlet and Neumann bounds for disks and balls using refined lattice-point/trapezoid-sum techniques, and derives detailed, dimension-dependent cylinder bounds. A key technical innovation is leveraging higher-order information about the auxiliary function g and its derivatives, enabling tighter control of eigenvalue counts. The results deepen our understanding of how geometry constrains Laplacian spectra, and they extend verifications of Pólya’s conjecture to new, geometrically nontrivial domains and product-domain settings.

Abstract

Pólya's conjecture on the eigenvalues of the Laplacian has been one of the core problems in spectral geometry. Building upon the recent breakthrough works on Pólya's conjecture for balls and annuli by Filonov, Levitin, Polterovich and Sher, we study several aspects of Pólya's conjecture for balls and cylinders: by refining the purely analytical portion of the proof in [2] for the Neumann Pólya's conjecture for the disk, we extend the regime of the spectral parameter that can be established without computer assistance; we obtain improvement of Pólya's conjecture for disks and balls; we obtain improvement of Pólya's conjecture for cylinders and confirm the Neumann Pólya's conjecture for cylinders in $\mathbb{R}^3$. As a supplementary effort, we study Weyl's law for cylinders.

Theorems & Definitions (34)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3: FLPS:2023
• Theorem 1.4
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10
• Corollary 1.11