{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/spherical-caps-do-not-always-maximize-neumann","title":"Spherical caps do not always maximize Neumann eigenvalues on the sphere","arxiv_id":"2503.15385","date":"2025-03-19","proceeding":null,"authors":["Dorin Bucur","Richard S. Laugesen","Eloi Martinet","Mickaël Nahon"],"abstract":"We prove the existence of an open set $\\Omega\\subset\\mathbb{S}^2$ for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.","url_abs":"https://arxiv.org/abs/2503.15385v1","url_pdf":"https://arxiv.org/pdf/2503.15385v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"spherical-caps-do-not-always-maximize-neumann","repo_url":"https://github.com/EloiMartinet/PuncturedSpheres","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}