{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:5UU2N32LGQYBLYBPC36372C5NS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"cd37c1db0568b0a708c626eca665e8487b616ffacdad3f08a4c34058de9828d9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2024-03-04T03:29:58Z","title_canon_sha256":"806241e21f8032ba82a22f22a07d7ec72b662990e0b7402da44f75fe780b4c0c"},"schema_version":"1.0","source":{"id":"2403.01701","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2403.01701","created_at":"2026-07-05T07:51:50Z"},{"alias_kind":"arxiv_version","alias_value":"2403.01701v1","created_at":"2026-07-05T07:51:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.01701","created_at":"2026-07-05T07:51:50Z"},{"alias_kind":"pith_short_12","alias_value":"5UU2N32LGQYB","created_at":"2026-07-05T07:51:50Z"},{"alias_kind":"pith_short_16","alias_value":"5UU2N32LGQYBLYBP","created_at":"2026-07-05T07:51:50Z"},{"alias_kind":"pith_short_8","alias_value":"5UU2N32L","created_at":"2026-07-05T07:51:50Z"}],"graph_snapshots":[{"event_id":"sha256:5946a92587e225edcab3ddad5822061185d4fdc6fb41ee8f972a03ad6e02f613","target":"graph","created_at":"2026-07-05T07:51:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2403.01701/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Carlos Pe\\~nafiel, Qing Cui","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2024-03-04T03:29:58Z","title":"A new characterization for Clifford hypersurfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.01701","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:af5f9c326dbde9ad0fd814025036456c70133884c1494f69289dae6924f43811","target":"record","created_at":"2026-07-05T07:51:50Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"cd37c1db0568b0a708c626eca665e8487b616ffacdad3f08a4c34058de9828d9","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.DG","submitted_at":"2024-03-04T03:29:58Z","title_canon_sha256":"806241e21f8032ba82a22f22a07d7ec72b662990e0b7402da44f75fe780b4c0c"},"schema_version":"1.0","source":{"id":"2403.01701","kind":"arxiv","version":1}},"canonical_sha256":"ed29a6ef4b343015e02f16fdbfe85d6c9322b108fe7a13ed57ae6eb980df01c5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ed29a6ef4b343015e02f16fdbfe85d6c9322b108fe7a13ed57ae6eb980df01c5","first_computed_at":"2026-07-05T07:51:50.603344Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:51:50.603344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/OarL8Up63Jh+V4UN7road9OnVgzHDUmux5bqFTFXTGHTzNI8O5AtjK9MOqhlfj+KmwKX89Zg1eyE8LuALvqBg==","signature_status":"signed_v1","signed_at":"2026-07-05T07:51:50.603700Z","signed_message":"canonical_sha256_bytes"},"source_id":"2403.01701","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:af5f9c326dbde9ad0fd814025036456c70133884c1494f69289dae6924f43811","sha256:5946a92587e225edcab3ddad5822061185d4fdc6fb41ee8f972a03ad6e02f613"],"state_sha256":"f63da2902da3d8337ffabde1fc681206ba1d68f537bf1317a92df7ea49fea322"}