{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:JGLFIKHD6AHXVRWYB7RJULNUQF","short_pith_number":"pith:JGLFIKHD","schema_version":"1.0","canonical_sha256":"49965428e3f00f7ac6d80fe29a2db48148217f9fa280d570dc5b57648c69a7f4","source":{"kind":"arxiv","id":"2408.10781","version":2},"attestation_state":"computed","paper":{"title":"$C^2$ estimates for $k$-Hessian equations and a rigidity theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Ruijia Zhang","submitted_at":"2024-08-20T12:20:27Z","abstract_excerpt":"We derive a concavity inequality for $k$-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the $k$-Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for $k$-curvature equations."},"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":"2408.10781","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2024-08-20T12:20:27Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"11c624b2d912547d7429c5e6fdeda325f79d1bbc9197fd323194b6a08ab131cd","abstract_canon_sha256":"bcafb7aeeb18d07b4210316b4d780564af6e801248ae5ecdcb2a058d7190aa51"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:14:44.794185Z","signature_b64":"0nMGY302Hh2OcMo99y/mj8R2zrD7qqUQobl2WDGdtUZxPlL577uI/iiia/sZXGfiplMPk3//bInpGs/YaAydCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49965428e3f00f7ac6d80fe29a2db48148217f9fa280d570dc5b57648c69a7f4","last_reissued_at":"2026-07-05T10:14:44.793684Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:14:44.793684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$C^2$ estimates for $k$-Hessian equations and a rigidity theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Ruijia Zhang","submitted_at":"2024-08-20T12:20:27Z","abstract_excerpt":"We derive a concavity inequality for $k$-Hessian operators under the semi-convexity condition. As an application, we establish interior estimates for semi-convex solutions of the $k$-Hessian equations with vanishing Dirichlet boundary and obtain a Liouville-type result. Additionally, we provide new and simple proofs of Guan-Ren-Wang's results on global curvature estimates for $k$-curvature equations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.10781","kind":"arxiv","version":2},"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/2408.10781/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":"2408.10781","created_at":"2026-07-05T10:14:44.793743+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.10781v2","created_at":"2026-07-05T10:14:44.793743+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.10781","created_at":"2026-07-05T10:14:44.793743+00:00"},{"alias_kind":"pith_short_12","alias_value":"JGLFIKHD6AHX","created_at":"2026-07-05T10:14:44.793743+00:00"},{"alias_kind":"pith_short_16","alias_value":"JGLFIKHD6AHXVRWY","created_at":"2026-07-05T10:14:44.793743+00:00"},{"alias_kind":"pith_short_8","alias_value":"JGLFIKHD","created_at":"2026-07-05T10:14:44.793743+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2501.01017","citing_title":"Second order estimates for $\\chi$-semi convex solutions of Hessian equations on Hermitian manifolds","ref_index":43,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF","json":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF.json","graph_json":"https://pith.science/api/pith-number/JGLFIKHD6AHXVRWYB7RJULNUQF/graph.json","events_json":"https://pith.science/api/pith-number/JGLFIKHD6AHXVRWYB7RJULNUQF/events.json","paper":"https://pith.science/paper/JGLFIKHD"},"agent_actions":{"view_html":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF","download_json":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF.json","view_paper":"https://pith.science/paper/JGLFIKHD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.10781&json=true","fetch_graph":"https://pith.science/api/pith-number/JGLFIKHD6AHXVRWYB7RJULNUQF/graph.json","fetch_events":"https://pith.science/api/pith-number/JGLFIKHD6AHXVRWYB7RJULNUQF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF/action/storage_attestation","attest_author":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF/action/author_attestation","sign_citation":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF/action/citation_signature","submit_replication":"https://pith.science/pith/JGLFIKHD6AHXVRWYB7RJULNUQF/action/replication_record"}},"created_at":"2026-07-05T10:14:44.793743+00:00","updated_at":"2026-07-05T10:14:44.793743+00:00"}