{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:IWA4C664OYJMI4F2FQO2WISVRY","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":"b5e6b405d3da7164572c412642f020e268f6b8a9954f29b2b831cce2d578d678","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-09-15T21:42:19Z","title_canon_sha256":"4c293b2ec3986c09267d8791422f636e575a58826ed91f9fc3f0e328dfefd6c7"},"schema_version":"1.0","source":{"id":"1409.4455","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.4455","created_at":"2026-05-18T01:17:57Z"},{"alias_kind":"arxiv_version","alias_value":"1409.4455v2","created_at":"2026-05-18T01:17:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.4455","created_at":"2026-05-18T01:17:57Z"},{"alias_kind":"pith_short_12","alias_value":"IWA4C664OYJM","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_16","alias_value":"IWA4C664OYJMI4F2","created_at":"2026-05-18T12:28:33Z"},{"alias_kind":"pith_short_8","alias_value":"IWA4C664","created_at":"2026-05-18T12:28:33Z"}],"graph_snapshots":[{"event_id":"sha256:fe45d5f94b55d0aa772f11c906a4ebd127a0a0d34285a8d4b106870b43a97536","target":"graph","created_at":"2026-05-18T01:17:57Z","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":"We propose a natural definition of the weighted $\\sigma_k$-curvature for a manifold with density; i.e.\\ a triple $(M^n,g,e^{-\\phi}\\mathrm{dvol})$. This definition is intended to capture the key properties of the $\\sigma_k$-curvatures in conformal geometry with the role of pointwise conformal changes of the metric replaced by pointwise changes of the measure. We justify our definition through three main results. First, we show that shrinking gradient Ricci solitons are local extrema of the total weighted $\\sigma_k$-curvature functionals when the weighted $\\sigma_k$-curvature is variational. Sec","authors_text":"Jeffrey S. Case","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-09-15T21:42:19Z","title":"A notion of the weighted $\\sigma_k$-curvature for manifolds with density"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.4455","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":"sha256:9261292188725282613c332d5b9aa648d777a224cffc024db23ecc77e418be78","target":"record","created_at":"2026-05-18T01:17:57Z","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":"b5e6b405d3da7164572c412642f020e268f6b8a9954f29b2b831cce2d578d678","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2014-09-15T21:42:19Z","title_canon_sha256":"4c293b2ec3986c09267d8791422f636e575a58826ed91f9fc3f0e328dfefd6c7"},"schema_version":"1.0","source":{"id":"1409.4455","kind":"arxiv","version":2}},"canonical_sha256":"4581c17bdc7612c470ba2c1dab22558e2035e5931028100083488e5458c3230e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4581c17bdc7612c470ba2c1dab22558e2035e5931028100083488e5458c3230e","first_computed_at":"2026-05-18T01:17:57.440841Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:57.440841Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/EvPl5DGtPVIAexb2Uf0CA5ysuVbnAiWwwa1tixYv6tTDxVJF5y6U1A8NMPq8d/MPz3zxJ67BOTHn1+R7tJ4Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:57.441535Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.4455","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9261292188725282613c332d5b9aa648d777a224cffc024db23ecc77e418be78","sha256:fe45d5f94b55d0aa772f11c906a4ebd127a0a0d34285a8d4b106870b43a97536"],"state_sha256":"d929e996c8a187610f3c5eedfda02848881d1d2fbaa5e2154669e426aa2d23c0"}