{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:M63OBXAD43T6JI6L3ZZFZTA6MO","short_pith_number":"pith:M63OBXAD","schema_version":"1.0","canonical_sha256":"67b6e0dc03e6e7e4a3cbde725ccc1e6394d6107a2d7ab5a8a628ea575a28bdc6","source":{"kind":"arxiv","id":"2607.06951","version":1},"attestation_state":"computed","paper":{"title":"A sharp isoperimetric inequality and the top order $Q$-curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Mingxiang Li, Xingwang Xu","submitted_at":"2026-07-08T03:26:19Z","abstract_excerpt":"For a smooth, complete and normal metric $g = e^{2u}|dx|^2$ with finite total $n$-th order $Q$-curvature on $\\mathbb{R}^n$ with dimension $n \\geq 2$, we first show that everywhere non-negativity (resp. non-positivity) $n$-th order $Q$-curvature $Q_g^{(n)}$ implies everywhere non-negativity (resp. non-positivity) of the sectional curvature. Based on this fact, we secondly show that, once $Q_g^{(n)}$ is non-negative, then for any compact domain $\\Omega \\subset \\mathbb{R}^n$ with smooth boundary $\\partial\\Omega$, the following sharp isoperimetric inequality holds: $$|\\partial\\Omega|_g^{\\frac{n}{n"},"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":"2607.06951","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-07-08T03:26:19Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"e360907098905e0761c227f9840ee6ba82587116897b3f42c3200ae80569b1f8","abstract_canon_sha256":"8e22268c7f94dce4ea41537caaa2289ff782415bfab07bc601a24dbf14a4a159"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-09T00:19:40.974040Z","signature_b64":"POV34SvsC48Fa7dFM320X9TWCiRexH67YSWmvi/0e10JgdMoppnMQgvoPFMK0zQq1dgbDqWhUSN7MiiM6eZ0Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67b6e0dc03e6e7e4a3cbde725ccc1e6394d6107a2d7ab5a8a628ea575a28bdc6","last_reissued_at":"2026-07-09T00:19:40.973626Z","signature_status":"signed_v1","first_computed_at":"2026-07-09T00:19:40.973626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A sharp isoperimetric inequality and the top order $Q$-curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Mingxiang Li, Xingwang Xu","submitted_at":"2026-07-08T03:26:19Z","abstract_excerpt":"For a smooth, complete and normal metric $g = e^{2u}|dx|^2$ with finite total $n$-th order $Q$-curvature on $\\mathbb{R}^n$ with dimension $n \\geq 2$, we first show that everywhere non-negativity (resp. non-positivity) $n$-th order $Q$-curvature $Q_g^{(n)}$ implies everywhere non-negativity (resp. non-positivity) of the sectional curvature. Based on this fact, we secondly show that, once $Q_g^{(n)}$ is non-negative, then for any compact domain $\\Omega \\subset \\mathbb{R}^n$ with smooth boundary $\\partial\\Omega$, the following sharp isoperimetric inequality holds: $$|\\partial\\Omega|_g^{\\frac{n}{n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.06951","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/2607.06951/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.06951","created_at":"2026-07-09T00:19:40.973691+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.06951v1","created_at":"2026-07-09T00:19:40.973691+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.06951","created_at":"2026-07-09T00:19:40.973691+00:00"},{"alias_kind":"pith_short_12","alias_value":"M63OBXAD43T6","created_at":"2026-07-09T00:19:40.973691+00:00"},{"alias_kind":"pith_short_16","alias_value":"M63OBXAD43T6JI6L","created_at":"2026-07-09T00:19:40.973691+00:00"},{"alias_kind":"pith_short_8","alias_value":"M63OBXAD","created_at":"2026-07-09T00:19:40.973691+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/M63OBXAD43T6JI6L3ZZFZTA6MO","json":"https://pith.science/pith/M63OBXAD43T6JI6L3ZZFZTA6MO.json","graph_json":"https://pith.science/api/pith-number/M63OBXAD43T6JI6L3ZZFZTA6MO/graph.json","events_json":"https://pith.science/api/pith-number/M63OBXAD43T6JI6L3ZZFZTA6MO/events.json","paper":"https://pith.science/paper/M63OBXAD"},"agent_actions":{"view_html":"https://pith.science/pith/M63OBXAD43T6JI6L3ZZFZTA6MO","download_json":"https://pith.science/pith/M63OBXAD43T6JI6L3ZZFZTA6MO.json","view_paper":"https://pith.science/paper/M63OBXAD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.06951&json=true","fetch_graph":"https://pith.science/api/pith-number/M63OBXAD43T6JI6L3ZZFZTA6MO/graph.json","fetch_events":"https://pith.science/api/pith-number/M63OBXAD43T6JI6L3ZZFZTA6MO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/M63OBXAD43T6JI6L3ZZFZTA6MO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/M63OBXAD43T6JI6L3ZZFZTA6MO/action/storage_attestation","attest_author":"https://pith.science/pith/M63OBXAD43T6JI6L3ZZFZTA6MO/action/author_attestation","sign_citation":"https://pith.science/pith/M63OBXAD43T6JI6L3ZZFZTA6MO/action/citation_signature","submit_replication":"https://pith.science/pith/M63OBXAD43T6JI6L3ZZFZTA6MO/action/replication_record"}},"created_at":"2026-07-09T00:19:40.973691+00:00","updated_at":"2026-07-09T00:19:40.973691+00:00"}