{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:XCPBYDH7Y6R7ENH3IRFYWNSXLE","short_pith_number":"pith:XCPBYDH7","schema_version":"1.0","canonical_sha256":"b89e1c0cffc7a3f234fb444b8b3657590f9c31a0caa9ee066c14c742f51b6836","source":{"kind":"arxiv","id":"1608.01663","version":1},"attestation_state":"computed","paper":{"title":"The weighted $\\sigma_k$-curvature of a smooth metric measure space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Jeffrey S. Case","submitted_at":"2016-08-04T19:56:10Z","abstract_excerpt":"We propose a definition of the weighted $\\sigma_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $\\sigma_k$-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when $k=1,2$ or the smooth metric measure space is locally conformally flat in the weighted sense. Second, we show that, in the variational cases, quasi-Einstein metrics are stable with respect to the total weighted $\\sigma_k$-curvature functional. We also discuss related conjectures for weighted Einstein manifolds."},"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":"1608.01663","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-08-04T19:56:10Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"47a91ede57168c62d7179f391e876f3d29d981b78f7344a4c2e5f12e30dbadbd","abstract_canon_sha256":"47127d56123fb98891c22e298d9c05c9ebcd43c472dbf7a33701a2b4f6fa60f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:15.758483Z","signature_b64":"o517aRVyaarAr4on7m6hDwADHZF8BOfZUEwQOR8dLdl0Y/H1WqOi8gH4iK50TOt2Y6atkXO44iyZG5Sslqi/DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b89e1c0cffc7a3f234fb444b8b3657590f9c31a0caa9ee066c14c742f51b6836","last_reissued_at":"2026-05-17T23:44:15.757728Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:15.757728Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The weighted $\\sigma_k$-curvature of a smooth metric measure space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Jeffrey S. Case","submitted_at":"2016-08-04T19:56:10Z","abstract_excerpt":"We propose a definition of the weighted $\\sigma_k$-curvature of a smooth metric measure space and justify it in two ways. First, we show that the weighted $\\sigma_k$-curvature prescription problem is governed by a fully nonlinear second order elliptic PDE which is variational when $k=1,2$ or the smooth metric measure space is locally conformally flat in the weighted sense. Second, we show that, in the variational cases, quasi-Einstein metrics are stable with respect to the total weighted $\\sigma_k$-curvature functional. We also discuss related conjectures for weighted Einstein manifolds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01663","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":"1608.01663","created_at":"2026-05-17T23:44:15.757856+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.01663v1","created_at":"2026-05-17T23:44:15.757856+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01663","created_at":"2026-05-17T23:44:15.757856+00:00"},{"alias_kind":"pith_short_12","alias_value":"XCPBYDH7Y6R7","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"XCPBYDH7Y6R7ENH3","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"XCPBYDH7","created_at":"2026-05-18T12:30:51.357362+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/XCPBYDH7Y6R7ENH3IRFYWNSXLE","json":"https://pith.science/pith/XCPBYDH7Y6R7ENH3IRFYWNSXLE.json","graph_json":"https://pith.science/api/pith-number/XCPBYDH7Y6R7ENH3IRFYWNSXLE/graph.json","events_json":"https://pith.science/api/pith-number/XCPBYDH7Y6R7ENH3IRFYWNSXLE/events.json","paper":"https://pith.science/paper/XCPBYDH7"},"agent_actions":{"view_html":"https://pith.science/pith/XCPBYDH7Y6R7ENH3IRFYWNSXLE","download_json":"https://pith.science/pith/XCPBYDH7Y6R7ENH3IRFYWNSXLE.json","view_paper":"https://pith.science/paper/XCPBYDH7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.01663&json=true","fetch_graph":"https://pith.science/api/pith-number/XCPBYDH7Y6R7ENH3IRFYWNSXLE/graph.json","fetch_events":"https://pith.science/api/pith-number/XCPBYDH7Y6R7ENH3IRFYWNSXLE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XCPBYDH7Y6R7ENH3IRFYWNSXLE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XCPBYDH7Y6R7ENH3IRFYWNSXLE/action/storage_attestation","attest_author":"https://pith.science/pith/XCPBYDH7Y6R7ENH3IRFYWNSXLE/action/author_attestation","sign_citation":"https://pith.science/pith/XCPBYDH7Y6R7ENH3IRFYWNSXLE/action/citation_signature","submit_replication":"https://pith.science/pith/XCPBYDH7Y6R7ENH3IRFYWNSXLE/action/replication_record"}},"created_at":"2026-05-17T23:44:15.757856+00:00","updated_at":"2026-05-17T23:44:15.757856+00:00"}