Improvement of Pólya's conjecture for balls and cylinders
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909936657629184 |
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| author | Guo, Jingwei Miao, Changxing Wang, Weiwei Zhan, Guoqing |
| author_facet | Guo, Jingwei Miao, Changxing Wang, Weiwei Zhan, Guoqing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_17050 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improvement of Pólya's conjecture for balls and cylinders Guo, Jingwei Miao, Changxing Wang, Weiwei Zhan, Guoqing Classical Analysis and ODEs Spectral Theory 35P15, 35P20 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. |
| title | Improvement of Pólya's conjecture for balls and cylinders |
| topic | Classical Analysis and ODEs Spectral Theory 35P15, 35P20 |
| url | https://arxiv.org/abs/2511.17050 |