{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:6QSXPFQ4RZZJWYQC43W3VF55CA","short_pith_number":"pith:6QSXPFQ4","canonical_record":{"source":{"id":"1110.3557","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-10-17T02:11:23Z","cross_cats_sorted":[],"title_canon_sha256":"bceb4cc64db54664050c0d1a8fdd4c8c36f3eb17e458b97affec11393914d943","abstract_canon_sha256":"f5f10fe2aed7fb992d32aa3e93c6a8f7c8f1ded2342cefd0de383e4beaf4d976"},"schema_version":"1.0"},"canonical_sha256":"f42577961c8e729b6202e6edba97bd100b1a236d9025e5b52a08828dc35eebba","source":{"kind":"arxiv","id":"1110.3557","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.3557","created_at":"2026-05-18T03:59:49Z"},{"alias_kind":"arxiv_version","alias_value":"1110.3557v2","created_at":"2026-05-18T03:59:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.3557","created_at":"2026-05-18T03:59:49Z"},{"alias_kind":"pith_short_12","alias_value":"6QSXPFQ4RZZJ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6QSXPFQ4RZZJWYQC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6QSXPFQ4","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:6QSXPFQ4RZZJWYQC43W3VF55CA","target":"record","payload":{"canonical_record":{"source":{"id":"1110.3557","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-10-17T02:11:23Z","cross_cats_sorted":[],"title_canon_sha256":"bceb4cc64db54664050c0d1a8fdd4c8c36f3eb17e458b97affec11393914d943","abstract_canon_sha256":"f5f10fe2aed7fb992d32aa3e93c6a8f7c8f1ded2342cefd0de383e4beaf4d976"},"schema_version":"1.0"},"canonical_sha256":"f42577961c8e729b6202e6edba97bd100b1a236d9025e5b52a08828dc35eebba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:49.354727Z","signature_b64":"TivN1Ee2WBRA7TPAZEu3hkGnM2n5PLFn11n/VGIdnyCIad+ODk+i6EWA18uLrzjmMtgPRJU9r+VgCsh1swmiCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f42577961c8e729b6202e6edba97bd100b1a236d9025e5b52a08828dc35eebba","last_reissued_at":"2026-05-18T03:59:49.354069Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:49.354069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1110.3557","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:59:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hVZ0xP1AjhaZ4A2Z6YZjG5na0we4AFWQT8HMRJ7yuUFy/NbHS3OEI4x2qRQ+OBkkS3kw6+jW25uLODMsyBUHCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T01:56:09.792908Z"},"content_sha256":"b05537a21621a92cf55fc17d9822a17c6ccccac871b9f15c2c5ff9f9c6725996","schema_version":"1.0","event_id":"sha256:b05537a21621a92cf55fc17d9822a17c6ccccac871b9f15c2c5ff9f9c6725996"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:6QSXPFQ4RZZJWYQC43W3VF55CA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"New examples of Willmore submanifolds in the unit sphere via isoparametric functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Wenjiao Yan, Zizhou Tang","submitted_at":"2011-10-17T02:11:23Z","abstract_excerpt":"An isometric immersion $x:M^n\\rightarrow S^{n+p}$ is called Willmore if it is an extremal submanifold of the Willmore functional: $W(x)=\\int_{M^n} (S-nH^2)^{\\frac{n}{2}}dv$, where $S$ is the norm square of the second fundamental form and $H$ is the mean curvature. Examples of Willmore submanifolds in the unit sphere are scarce in the literature. The present paper gives a series of new examples of Willmore submanifolds in the unit sphere via isoparametric functions of FKM-type."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3557","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:59:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5gNvGI+OCTlDwLEesQCB+t2COAloOXChxqrJdbgrJYi2RccWaWhVouqg0ofLjFk9Hm03pAYaW9wY3MLgYHvUCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T01:56:09.793252Z"},"content_sha256":"31b8f32e06b46500e3f649afd0d419ed2d55a594670add47bf541af00d3ae71f","schema_version":"1.0","event_id":"sha256:31b8f32e06b46500e3f649afd0d419ed2d55a594670add47bf541af00d3ae71f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6QSXPFQ4RZZJWYQC43W3VF55CA/bundle.json","state_url":"https://pith.science/pith/6QSXPFQ4RZZJWYQC43W3VF55CA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6QSXPFQ4RZZJWYQC43W3VF55CA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T01:56:09Z","links":{"resolver":"https://pith.science/pith/6QSXPFQ4RZZJWYQC43W3VF55CA","bundle":"https://pith.science/pith/6QSXPFQ4RZZJWYQC43W3VF55CA/bundle.json","state":"https://pith.science/pith/6QSXPFQ4RZZJWYQC43W3VF55CA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6QSXPFQ4RZZJWYQC43W3VF55CA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:6QSXPFQ4RZZJWYQC43W3VF55CA","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":"f5f10fe2aed7fb992d32aa3e93c6a8f7c8f1ded2342cefd0de383e4beaf4d976","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-10-17T02:11:23Z","title_canon_sha256":"bceb4cc64db54664050c0d1a8fdd4c8c36f3eb17e458b97affec11393914d943"},"schema_version":"1.0","source":{"id":"1110.3557","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.3557","created_at":"2026-05-18T03:59:49Z"},{"alias_kind":"arxiv_version","alias_value":"1110.3557v2","created_at":"2026-05-18T03:59:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.3557","created_at":"2026-05-18T03:59:49Z"},{"alias_kind":"pith_short_12","alias_value":"6QSXPFQ4RZZJ","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"6QSXPFQ4RZZJWYQC","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"6QSXPFQ4","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:31b8f32e06b46500e3f649afd0d419ed2d55a594670add47bf541af00d3ae71f","target":"graph","created_at":"2026-05-18T03:59:49Z","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"},"paper":{"abstract_excerpt":"An isometric immersion $x:M^n\\rightarrow S^{n+p}$ is called Willmore if it is an extremal submanifold of the Willmore functional: $W(x)=\\int_{M^n} (S-nH^2)^{\\frac{n}{2}}dv$, where $S$ is the norm square of the second fundamental form and $H$ is the mean curvature. Examples of Willmore submanifolds in the unit sphere are scarce in the literature. The present paper gives a series of new examples of Willmore submanifolds in the unit sphere via isoparametric functions of FKM-type.","authors_text":"Wenjiao Yan, Zizhou Tang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-10-17T02:11:23Z","title":"New examples of Willmore submanifolds in the unit sphere via isoparametric functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3557","kind":"arxiv","version":2},"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:b05537a21621a92cf55fc17d9822a17c6ccccac871b9f15c2c5ff9f9c6725996","target":"record","created_at":"2026-05-18T03:59:49Z","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":"f5f10fe2aed7fb992d32aa3e93c6a8f7c8f1ded2342cefd0de383e4beaf4d976","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-10-17T02:11:23Z","title_canon_sha256":"bceb4cc64db54664050c0d1a8fdd4c8c36f3eb17e458b97affec11393914d943"},"schema_version":"1.0","source":{"id":"1110.3557","kind":"arxiv","version":2}},"canonical_sha256":"f42577961c8e729b6202e6edba97bd100b1a236d9025e5b52a08828dc35eebba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f42577961c8e729b6202e6edba97bd100b1a236d9025e5b52a08828dc35eebba","first_computed_at":"2026-05-18T03:59:49.354069Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:59:49.354069Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TivN1Ee2WBRA7TPAZEu3hkGnM2n5PLFn11n/VGIdnyCIad+ODk+i6EWA18uLrzjmMtgPRJU9r+VgCsh1swmiCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:59:49.354727Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.3557","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b05537a21621a92cf55fc17d9822a17c6ccccac871b9f15c2c5ff9f9c6725996","sha256:31b8f32e06b46500e3f649afd0d419ed2d55a594670add47bf541af00d3ae71f"],"state_sha256":"955a8a813511bccd25490a015b14f5c4442757afe4f598f575987aa3285ddab5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GeylTnKMUIOKtgN8wEV3wcYrkwtlFDTR/aVVbnmU1bI5tRwCdumuhAB6P8sJkDZatO9f7yK8CmiJDNxwd4SmCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T01:56:09.795175Z","bundle_sha256":"a1e128399ae9496f6266667eb4ecf41630c1e93bad761b1c80d64c8275566186"}}