{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:UTIUBQUJZYGX4M3OOHMREJXHH5","short_pith_number":"pith:UTIUBQUJ","canonical_record":{"source":{"id":"2106.04504","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-06-08T16:44:39Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"a2b6414cbb9763cf2768f11e0fad00c4b19ce450077e3793429c690e8689c8ce","abstract_canon_sha256":"f7221fdd6f61451dd3959d7cf382355aca2ee90969e3fc44ae4e10dc96b15475"},"schema_version":"1.0"},"canonical_sha256":"a4d140c289ce0d7e336e71d91226e73f430126c2fb31c3f54a2f022344168e89","source":{"kind":"arxiv","id":"2106.04504","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.04504","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"arxiv_version","alias_value":"2106.04504v1","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.04504","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"pith_short_12","alias_value":"UTIUBQUJZYGX","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"pith_short_16","alias_value":"UTIUBQUJZYGX4M3O","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"pith_short_8","alias_value":"UTIUBQUJ","created_at":"2026-07-05T02:47:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:UTIUBQUJZYGX4M3OOHMREJXHH5","target":"record","payload":{"canonical_record":{"source":{"id":"2106.04504","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-06-08T16:44:39Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"a2b6414cbb9763cf2768f11e0fad00c4b19ce450077e3793429c690e8689c8ce","abstract_canon_sha256":"f7221fdd6f61451dd3959d7cf382355aca2ee90969e3fc44ae4e10dc96b15475"},"schema_version":"1.0"},"canonical_sha256":"a4d140c289ce0d7e336e71d91226e73f430126c2fb31c3f54a2f022344168e89","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:47:15.959928Z","signature_b64":"lezFAxKYm5qcpRjv4N4SheiH2v3T0f1aZBW/6FUGTeX8XAJIIHPfnbnty04O01NtFlmfPN8CEEvOCpDkMOIYBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a4d140c289ce0d7e336e71d91226e73f430126c2fb31c3f54a2f022344168e89","last_reissued_at":"2026-07-05T02:47:15.959542Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:47:15.959542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2106.04504","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:47:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ja68sY6ByDOMUZFIgALT0b0qSLq9x6rB87YroxyTCTixsIRhVSxKQkttH77lrIqesdMs2Ub3XkdLUtU6eIw2DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-31T18:18:10.672087Z"},"content_sha256":"47a9cf89cbb3b85aad4c0aad54b62f8694c66936ba5bc12d1a9bc261aa5c75e6","schema_version":"1.0","event_id":"sha256:47a9cf89cbb3b85aad4c0aad54b62f8694c66936ba5bc12d1a9bc261aa5c75e6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:UTIUBQUJZYGX4M3OOHMREJXHH5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The axisymmetric $\\sigma_k$-Nirenberg problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Bo Wang, Luc Nguyen, Yanyan Li","submitted_at":"2021-06-08T16:44:39Z","abstract_excerpt":"We study the problem of prescribing $\\sigma_k$-curvature for a conformal metric on the standard sphere $\\mathbb{S}^n$ with $2 \\leq k < n/2$ and $n \\geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $\\beta \\in [2,n)$. We prove among other things the following statements. (1) When $\\beta>n-2k$, the solution set is compact, has a no"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.04504","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/2106.04504/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:47:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3uI+ymcGERTPzac/A8hiuPkHp51vwHHK1PNSSYCR3PJITmJQFAC32K+6ay0GlLFKTS4+TMMUHFkdAxbYuYbLBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-31T18:18:10.673213Z"},"content_sha256":"73dcec657e31c3e3ca7cd9557ef2b6f8bad4bc8a564e8ca2562e06591651cc9b","schema_version":"1.0","event_id":"sha256:73dcec657e31c3e3ca7cd9557ef2b6f8bad4bc8a564e8ca2562e06591651cc9b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UTIUBQUJZYGX4M3OOHMREJXHH5/bundle.json","state_url":"https://pith.science/pith/UTIUBQUJZYGX4M3OOHMREJXHH5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UTIUBQUJZYGX4M3OOHMREJXHH5/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":"2026-07-31T18:18:10Z","links":{"resolver":"https://pith.science/pith/UTIUBQUJZYGX4M3OOHMREJXHH5","bundle":"https://pith.science/pith/UTIUBQUJZYGX4M3OOHMREJXHH5/bundle.json","state":"https://pith.science/pith/UTIUBQUJZYGX4M3OOHMREJXHH5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UTIUBQUJZYGX4M3OOHMREJXHH5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:UTIUBQUJZYGX4M3OOHMREJXHH5","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":"f7221fdd6f61451dd3959d7cf382355aca2ee90969e3fc44ae4e10dc96b15475","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-06-08T16:44:39Z","title_canon_sha256":"a2b6414cbb9763cf2768f11e0fad00c4b19ce450077e3793429c690e8689c8ce"},"schema_version":"1.0","source":{"id":"2106.04504","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2106.04504","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"arxiv_version","alias_value":"2106.04504v1","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2106.04504","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"pith_short_12","alias_value":"UTIUBQUJZYGX","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"pith_short_16","alias_value":"UTIUBQUJZYGX4M3O","created_at":"2026-07-05T02:47:15Z"},{"alias_kind":"pith_short_8","alias_value":"UTIUBQUJ","created_at":"2026-07-05T02:47:15Z"}],"graph_snapshots":[{"event_id":"sha256:73dcec657e31c3e3ca7cd9557ef2b6f8bad4bc8a564e8ca2562e06591651cc9b","target":"graph","created_at":"2026-07-05T02:47:15Z","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/2106.04504/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study the problem of prescribing $\\sigma_k$-curvature for a conformal metric on the standard sphere $\\mathbb{S}^n$ with $2 \\leq k < n/2$ and $n \\geq 5$ in axisymmetry. Compactness, non-compactness, existence and non-existence results are proved in terms of the behaviors of the prescribed curvature function $K$ near the north and the south poles. For example, consider the case when the north and the south poles are local maximum points of $K$ of flatness order $\\beta \\in [2,n)$. We prove among other things the following statements. (1) When $\\beta>n-2k$, the solution set is compact, has a no","authors_text":"Bo Wang, Luc Nguyen, Yanyan Li","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-06-08T16:44:39Z","title":"The axisymmetric $\\sigma_k$-Nirenberg problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.04504","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":"sha256:47a9cf89cbb3b85aad4c0aad54b62f8694c66936ba5bc12d1a9bc261aa5c75e6","target":"record","created_at":"2026-07-05T02:47:15Z","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":"f7221fdd6f61451dd3959d7cf382355aca2ee90969e3fc44ae4e10dc96b15475","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2021-06-08T16:44:39Z","title_canon_sha256":"a2b6414cbb9763cf2768f11e0fad00c4b19ce450077e3793429c690e8689c8ce"},"schema_version":"1.0","source":{"id":"2106.04504","kind":"arxiv","version":1}},"canonical_sha256":"a4d140c289ce0d7e336e71d91226e73f430126c2fb31c3f54a2f022344168e89","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a4d140c289ce0d7e336e71d91226e73f430126c2fb31c3f54a2f022344168e89","first_computed_at":"2026-07-05T02:47:15.959542Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:47:15.959542Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"lezFAxKYm5qcpRjv4N4SheiH2v3T0f1aZBW/6FUGTeX8XAJIIHPfnbnty04O01NtFlmfPN8CEEvOCpDkMOIYBw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:47:15.959928Z","signed_message":"canonical_sha256_bytes"},"source_id":"2106.04504","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:47a9cf89cbb3b85aad4c0aad54b62f8694c66936ba5bc12d1a9bc261aa5c75e6","sha256:73dcec657e31c3e3ca7cd9557ef2b6f8bad4bc8a564e8ca2562e06591651cc9b"],"state_sha256":"e482e75794ce114bde423b2a4efbcc1ad617e50afb50401e3425cd47aba89870"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GB2UQx59aiwzFqT49GvDb1W+kkUHSLRVhE8bsvc/YqAFClfjzob894sZNISI9La6bsGxWvYhLaeP8w6uQykbCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-31T18:18:10.681519Z","bundle_sha256":"ac84e732f32487e69d27d590e4ace640ea60c8b006f0ca2c302c3f8d68b097d1"}}