{"paper":{"title":"Eigenvalues on spheres","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.DG","authors_text":"Guoyi Xu, Haibin Wang, Shengjie Lin","submitted_at":"2026-07-13T13:32:25Z","abstract_excerpt":"For every smooth Riemannian metric on the two sphere whose Gaussian curvature is bounded below by one, we prove that each positive Laplace eigenvalue, counted with multiplicity, is no smaller than the corresponding eigenvalue of the unit round sphere. Equality at any positive position in the ordered spectrum forces the metric to be isometric to the unit round metric.\n  We further establish a sharp finite spectral counting comparison for Alexandrov two spheres with curvature bounded below by one. At every positive spectral threshold of the unit round sphere, the number of Laplace eigenvalues be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.11544","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.11544/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}