{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:YL3QLM2DNADFCXBYV6A7IYUTAH","short_pith_number":"pith:YL3QLM2D","schema_version":"1.0","canonical_sha256":"c2f705b3436806515c38af81f4629301db9788ce0953eafd5a96ce4ce68ba68b","source":{"kind":"arxiv","id":"1512.05389","version":1},"attestation_state":"computed","paper":{"title":"Deformations of Q-curvature I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Wei Yuan, Yueh-Ju Lin","submitted_at":"2015-12-16T21:55:43Z","abstract_excerpt":"In this article, we investigate deformation problems of $Q$-curvature on closed Riemannian manifolds. One of the most crucial notions we use is the $Q$-singular space, which was introduced by Chang-Gursky-Yang during 1990's. Inspired by the early work of Fischer-Marsden, we derived several results about geometry related to $Q$-curvature. It includes classifications for nonnegative Einstein $Q$-singular spaces, linearized stability of non-$Q$-singular spaces and a local rigidity result for flat manifolds with nonnegative $Q$-curvature. As for global results, we showed that any smooth function c"},"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":"1512.05389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-12-16T21:55:43Z","cross_cats_sorted":[],"title_canon_sha256":"71c81b8b2e7724534f4f34e0c433c276bce43460e6f61394a3af56d7e521e5ba","abstract_canon_sha256":"5879e93c01248d54f946b89ecd60bb6c16595c5d8094358a74c386bf8b833497"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:10.158711Z","signature_b64":"o1j5PhhlHW2F1lxDAvMeIXoW9oaBXiX3vvkyI5SYo9N0soj5mhYGF5UHtvnthqFSmkpOMGW2vMXcvRg+mriyDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c2f705b3436806515c38af81f4629301db9788ce0953eafd5a96ce4ce68ba68b","last_reissued_at":"2026-05-18T01:24:10.158113Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:10.158113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Deformations of Q-curvature I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Wei Yuan, Yueh-Ju Lin","submitted_at":"2015-12-16T21:55:43Z","abstract_excerpt":"In this article, we investigate deformation problems of $Q$-curvature on closed Riemannian manifolds. One of the most crucial notions we use is the $Q$-singular space, which was introduced by Chang-Gursky-Yang during 1990's. Inspired by the early work of Fischer-Marsden, we derived several results about geometry related to $Q$-curvature. It includes classifications for nonnegative Einstein $Q$-singular spaces, linearized stability of non-$Q$-singular spaces and a local rigidity result for flat manifolds with nonnegative $Q$-curvature. As for global results, we showed that any smooth function c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.05389","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":""},"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":"1512.05389","created_at":"2026-05-18T01:24:10.158205+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.05389v1","created_at":"2026-05-18T01:24:10.158205+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.05389","created_at":"2026-05-18T01:24:10.158205+00:00"},{"alias_kind":"pith_short_12","alias_value":"YL3QLM2DNADF","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_16","alias_value":"YL3QLM2DNADFCXBY","created_at":"2026-05-18T12:29:50.041715+00:00"},{"alias_kind":"pith_short_8","alias_value":"YL3QLM2D","created_at":"2026-05-18T12:29:50.041715+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/YL3QLM2DNADFCXBYV6A7IYUTAH","json":"https://pith.science/pith/YL3QLM2DNADFCXBYV6A7IYUTAH.json","graph_json":"https://pith.science/api/pith-number/YL3QLM2DNADFCXBYV6A7IYUTAH/graph.json","events_json":"https://pith.science/api/pith-number/YL3QLM2DNADFCXBYV6A7IYUTAH/events.json","paper":"https://pith.science/paper/YL3QLM2D"},"agent_actions":{"view_html":"https://pith.science/pith/YL3QLM2DNADFCXBYV6A7IYUTAH","download_json":"https://pith.science/pith/YL3QLM2DNADFCXBYV6A7IYUTAH.json","view_paper":"https://pith.science/paper/YL3QLM2D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.05389&json=true","fetch_graph":"https://pith.science/api/pith-number/YL3QLM2DNADFCXBYV6A7IYUTAH/graph.json","fetch_events":"https://pith.science/api/pith-number/YL3QLM2DNADFCXBYV6A7IYUTAH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YL3QLM2DNADFCXBYV6A7IYUTAH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YL3QLM2DNADFCXBYV6A7IYUTAH/action/storage_attestation","attest_author":"https://pith.science/pith/YL3QLM2DNADFCXBYV6A7IYUTAH/action/author_attestation","sign_citation":"https://pith.science/pith/YL3QLM2DNADFCXBYV6A7IYUTAH/action/citation_signature","submit_replication":"https://pith.science/pith/YL3QLM2DNADFCXBYV6A7IYUTAH/action/replication_record"}},"created_at":"2026-05-18T01:24:10.158205+00:00","updated_at":"2026-05-18T01:24:10.158205+00:00"}