{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:XX3TZNWHK5EUIBVBEY2WDIJX4F","short_pith_number":"pith:XX3TZNWH","schema_version":"1.0","canonical_sha256":"bdf73cb6c757494406a1263561a137e152939459ab9d1791761d5f3f9f83881f","source":{"kind":"arxiv","id":"1111.2650","version":3},"attestation_state":"computed","paper":{"title":"Variational formulas of higher order mean curvatures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jianquan Ge, Ling Xu","submitted_at":"2011-11-11T02:47:35Z","abstract_excerpt":"In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total $2p$-th mean curvature functional $\\mathcal {M}_{2p}$ of a submanifold $M^n$ in a general Riemannian manifold $N^{n+m}$ for $p=0,1,...,[\\frac{n}{2}]$. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional $\\mathcal {M}_{2p}$, called relatively $2p$-minimal submanifolds, for all $p$. At last, we discuss the relations between relatively $2p$-minimal submanifolds and austere submanifolds in real space forms, as well as a spec"},"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":"1111.2650","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2011-11-11T02:47:35Z","cross_cats_sorted":[],"title_canon_sha256":"850ae765be75ae2cfa2373f1474d86800a32f4ea5e5c7d3b41bc1d0e5ccc006b","abstract_canon_sha256":"1cb146c711af9633c8a75c43c86c522c58bd5495c7576c903b8ac7ab79c3dd28"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:42.182780Z","signature_b64":"OdVaU62YnH8XC4Hhd1HILlnMWoQkR5kiK22pCvL99s11lUSWKGDj/nENlLtf+J/8oQi2dpXvMQTbWuUqtQ66BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bdf73cb6c757494406a1263561a137e152939459ab9d1791761d5f3f9f83881f","last_reissued_at":"2026-05-18T01:59:42.182134Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:42.182134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Variational formulas of higher order mean curvatures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jianquan Ge, Ling Xu","submitted_at":"2011-11-11T02:47:35Z","abstract_excerpt":"In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total $2p$-th mean curvature functional $\\mathcal {M}_{2p}$ of a submanifold $M^n$ in a general Riemannian manifold $N^{n+m}$ for $p=0,1,...,[\\frac{n}{2}]$. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the functional $\\mathcal {M}_{2p}$, called relatively $2p$-minimal submanifolds, for all $p$. At last, we discuss the relations between relatively $2p$-minimal submanifolds and austere submanifolds in real space forms, as well as a spec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2650","kind":"arxiv","version":3},"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":"1111.2650","created_at":"2026-05-18T01:59:42.182211+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.2650v3","created_at":"2026-05-18T01:59:42.182211+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2650","created_at":"2026-05-18T01:59:42.182211+00:00"},{"alias_kind":"pith_short_12","alias_value":"XX3TZNWHK5EU","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"XX3TZNWHK5EUIBVB","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"XX3TZNWH","created_at":"2026-05-18T12:26:47.523578+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/XX3TZNWHK5EUIBVBEY2WDIJX4F","json":"https://pith.science/pith/XX3TZNWHK5EUIBVBEY2WDIJX4F.json","graph_json":"https://pith.science/api/pith-number/XX3TZNWHK5EUIBVBEY2WDIJX4F/graph.json","events_json":"https://pith.science/api/pith-number/XX3TZNWHK5EUIBVBEY2WDIJX4F/events.json","paper":"https://pith.science/paper/XX3TZNWH"},"agent_actions":{"view_html":"https://pith.science/pith/XX3TZNWHK5EUIBVBEY2WDIJX4F","download_json":"https://pith.science/pith/XX3TZNWHK5EUIBVBEY2WDIJX4F.json","view_paper":"https://pith.science/paper/XX3TZNWH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.2650&json=true","fetch_graph":"https://pith.science/api/pith-number/XX3TZNWHK5EUIBVBEY2WDIJX4F/graph.json","fetch_events":"https://pith.science/api/pith-number/XX3TZNWHK5EUIBVBEY2WDIJX4F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XX3TZNWHK5EUIBVBEY2WDIJX4F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XX3TZNWHK5EUIBVBEY2WDIJX4F/action/storage_attestation","attest_author":"https://pith.science/pith/XX3TZNWHK5EUIBVBEY2WDIJX4F/action/author_attestation","sign_citation":"https://pith.science/pith/XX3TZNWHK5EUIBVBEY2WDIJX4F/action/citation_signature","submit_replication":"https://pith.science/pith/XX3TZNWHK5EUIBVBEY2WDIJX4F/action/replication_record"}},"created_at":"2026-05-18T01:59:42.182211+00:00","updated_at":"2026-05-18T01:59:42.182211+00:00"}