{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:5UU2N32LGQYBLYBPC36372C5NS","short_pith_number":"pith:5UU2N32L","schema_version":"1.0","canonical_sha256":"ed29a6ef4b343015e02f16fdbfe85d6c9322b108fe7a13ed57ae6eb980df01c5","source":{"kind":"arxiv","id":"2403.01701","version":1},"attestation_state":"computed","paper":{"title":"A new characterization for Clifford hypersurfaces","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Carlos Pe\\~nafiel, Qing Cui","submitted_at":"2024-03-04T03:29:58Z","abstract_excerpt":"For a closed minimal immersed hypersurface $M$ in $\\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\\ge 2$, define a constant $\\sigma_k=\\dfrac{\\int_M (|A|^2)^k}{|M|}$. We show that $\\sigma_k \\ge 2^k$ provided $n=2$ and $M$ is not totally geodesic. When $n=4$ and $M$ has two distinct principal curvatures, we show $\\sigma_2 \\ge 16$. When $n\\ge 3$ and $M$ has two distinct principal curvatures, for each integer $k\\ge 2$, there exists a positive constant $\\delta_k(n)<n$, if $|A|^2\\ge \\delta_k(n)$, we have $\\sigma_k\\ge n^k$. All the equality holds iff $M$ is isometric to a Cliff"},"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":"2403.01701","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2024-03-04T03:29:58Z","cross_cats_sorted":[],"title_canon_sha256":"806241e21f8032ba82a22f22a07d7ec72b662990e0b7402da44f75fe780b4c0c","abstract_canon_sha256":"cd37c1db0568b0a708c626eca665e8487b616ffacdad3f08a4c34058de9828d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:51:50.603700Z","signature_b64":"/OarL8Up63Jh+V4UN7road9OnVgzHDUmux5bqFTFXTGHTzNI8O5AtjK9MOqhlfj+KmwKX89Zg1eyE8LuALvqBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ed29a6ef4b343015e02f16fdbfe85d6c9322b108fe7a13ed57ae6eb980df01c5","last_reissued_at":"2026-07-05T07:51:50.603344Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:51:50.603344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A new characterization for Clifford hypersurfaces","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Carlos Pe\\~nafiel, Qing Cui","submitted_at":"2024-03-04T03:29:58Z","abstract_excerpt":"For a closed minimal immersed hypersurface $M$ in $\\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\\ge 2$, define a constant $\\sigma_k=\\dfrac{\\int_M (|A|^2)^k}{|M|}$. We show that $\\sigma_k \\ge 2^k$ provided $n=2$ and $M$ is not totally geodesic. When $n=4$ and $M$ has two distinct principal curvatures, we show $\\sigma_2 \\ge 16$. When $n\\ge 3$ and $M$ has two distinct principal curvatures, for each integer $k\\ge 2$, there exists a positive constant $\\delta_k(n)<n$, if $|A|^2\\ge \\delta_k(n)$, we have $\\sigma_k\\ge n^k$. All the equality holds iff $M$ is isometric to a Cliff"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.01701","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/2403.01701/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2403.01701","created_at":"2026-07-05T07:51:50.603402+00:00"},{"alias_kind":"arxiv_version","alias_value":"2403.01701v1","created_at":"2026-07-05T07:51:50.603402+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.01701","created_at":"2026-07-05T07:51:50.603402+00:00"},{"alias_kind":"pith_short_12","alias_value":"5UU2N32LGQYB","created_at":"2026-07-05T07:51:50.603402+00:00"},{"alias_kind":"pith_short_16","alias_value":"5UU2N32LGQYBLYBP","created_at":"2026-07-05T07:51:50.603402+00:00"},{"alias_kind":"pith_short_8","alias_value":"5UU2N32L","created_at":"2026-07-05T07:51:50.603402+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS","json":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS.json","graph_json":"https://pith.science/api/pith-number/5UU2N32LGQYBLYBPC36372C5NS/graph.json","events_json":"https://pith.science/api/pith-number/5UU2N32LGQYBLYBPC36372C5NS/events.json","paper":"https://pith.science/paper/5UU2N32L"},"agent_actions":{"view_html":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS","download_json":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS.json","view_paper":"https://pith.science/paper/5UU2N32L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2403.01701&json=true","fetch_graph":"https://pith.science/api/pith-number/5UU2N32LGQYBLYBPC36372C5NS/graph.json","fetch_events":"https://pith.science/api/pith-number/5UU2N32LGQYBLYBPC36372C5NS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS/action/storage_attestation","attest_author":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS/action/author_attestation","sign_citation":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS/action/citation_signature","submit_replication":"https://pith.science/pith/5UU2N32LGQYBLYBPC36372C5NS/action/replication_record"}},"created_at":"2026-07-05T07:51:50.603402+00:00","updated_at":"2026-07-05T07:51:50.603402+00:00"}