{"paper":{"title":"A new gap for complete hypersurfaces with constant mean curvature in space forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hongwei Xu, Juanru Gu, Li Lei","submitted_at":"2018-10-31T02:40:07Z","abstract_excerpt":"Let $M$ be an $n$-dimensional closed hypersurface with constant mean curvature and constant scalar curvature in an unit sphere. Denote by $H$ and $S$ the mean curvature and the squared length of the second fundamental form respectively. We prove that if $S > \\alpha (n, H)$, where $n\\geq 4$ and $H\\neq 0$, then $S > \\alpha (n, H) + B_n\\frac{n H^2}{n - 1}$. Here \\[ \\alpha (n, H) = n + \\frac{n^3}{2 (n - 1)} H^2 - \\frac{n (n - 2)}{2 (n - 1)}\\sqrt{n^2 H^4 + 4 (n - 1) H^2}, \\] $B_n=\\frac{1}{5}$ for $4\\leq n \\leq 20$, and $B_n=\\frac{49}{250}$ for $n>20$. Moreover, we obtain a gap theorem for complete "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.13080","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":""},"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"}