{"paper":{"title":"Reciprocal sums of Neumann eigenvalues in non-Euclidean space forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Heng Zhang, Jiangcheng You","submitted_at":"2026-06-26T08:41:56Z","abstract_excerpt":"Let $M^n_\\kappa$ be the simply connected space form of dimension $n\\ge2$ and constant sectional curvature $\\kappa\\in\\{-1,1\\}$. For every bounded connected smooth domain $\\Omega\\subset M^n_\\kappa$, assume in the case $\\kappa=1$ that $\\Omega$ is contained in an open hemisphere, and let $B_\\Omega$ be a geodesic ball with $|B_\\Omega|=|\\Omega|$. We prove $$\n  \\sum_{j=1}^n \\frac1{\\mu_j(\\Omega)}\\ge \\frac{n}{\\mu_1(B_\\Omega)}, $$ where $\\mu_j(\\Omega)$ are the positive Neumann eigenvalues of $\\Omega$. Equality holds if and only if $\\Omega$ is a geodesic ball. This proves a conjecture proposed by Xia and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27848","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/2606.27848/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"}