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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1810.13080","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-10-31T02:40:07Z","cross_cats_sorted":[],"title_canon_sha256":"589b9d5f65a8a6a00159b127fc169eea46a4587205b5dbc9a2e5db761a4a7382","abstract_canon_sha256":"07dd89eba2fe9afe80e1e618d43011afd888d7912b882c8d267b13e5a547ff8a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:52.320125Z","signature_b64":"wtKyuxFr4BUKfgScwYY5ttQJ6jAle5F0XOO6vkjNTu+f2C85+/bVDnmcatBD/PTe2RS8JREHxAOB5mFhUMorDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a530476de9eaa233bc15f1ead3e6b1260b3035ecd6bb6e8e9eb310a3f935963","last_reissued_at":"2026-05-18T00:01:52.319696Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:52.319696Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $n>20$. 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