{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:3G47YG2P65QEZQHOUC2JJWMJBE","short_pith_number":"pith:3G47YG2P","schema_version":"1.0","canonical_sha256":"d9b9fc1b4ff7604cc0eea0b494d989093e639e7fcd2d4c5a3c744feea768ce36","source":{"kind":"arxiv","id":"1211.1474","version":2},"attestation_state":"computed","paper":{"title":"Uniqueness of generalized p-area minimizers and integrability of a horizontal normal in the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Jenn-Fang Hwang, Jih-Hsin Cheng","submitted_at":"2012-11-07T07:37:52Z","abstract_excerpt":"We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg group. The generalized $p$-area of a graph defined by $u$ reads $\\int |\\nabla u+\\vec{F}| + Hu$. If $u$ and $v$ are two minimizers for the generalized $p$-area satisfying the same Dirichlet boundary condition, then we can only get $N_{\\vec{F}}(u)$ $=$ $N_{\\vec{F}}(v)$ (on the nonsingular set) where $N_{\\vec{F}}(w)$ $:=$ $\\frac{\\nabla w+\\vec{F}}{|\\nabla w+\\vec{F}|}.$ To conclude $u$ $=$ $v$ (or $\\nabla u$ $=$ $\\nabla v)$, it is not straightforward as in the Riemannian case, but requires some special argument in general"},"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":"1211.1474","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-07T07:37:52Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"7642210c10a0e0635d734b14b15ad2d206aafef57aeff6e270ff1a10cb41917e","abstract_canon_sha256":"b3de72819cb96dcbca8b7c7f65f90a194a0335448cb2c3ea4de6906c914fec11"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:08.745940Z","signature_b64":"w5HPsk7H5Yutr6t3DKBRb0tilUMl9IsJvGc5Oth3Rp6VsNO/mm7Wk2oSsHngWYOY+iQyqlEB8eep05qBGLOQBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d9b9fc1b4ff7604cc0eea0b494d989093e639e7fcd2d4c5a3c744feea768ce36","last_reissued_at":"2026-05-18T03:26:08.745305Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:08.745305Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniqueness of generalized p-area minimizers and integrability of a horizontal normal in the Heisenberg group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Jenn-Fang Hwang, Jih-Hsin Cheng","submitted_at":"2012-11-07T07:37:52Z","abstract_excerpt":"We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg group. The generalized $p$-area of a graph defined by $u$ reads $\\int |\\nabla u+\\vec{F}| + Hu$. If $u$ and $v$ are two minimizers for the generalized $p$-area satisfying the same Dirichlet boundary condition, then we can only get $N_{\\vec{F}}(u)$ $=$ $N_{\\vec{F}}(v)$ (on the nonsingular set) where $N_{\\vec{F}}(w)$ $:=$ $\\frac{\\nabla w+\\vec{F}}{|\\nabla w+\\vec{F}|}.$ To conclude $u$ $=$ $v$ (or $\\nabla u$ $=$ $\\nabla v)$, it is not straightforward as in the Riemannian case, but requires some special argument in general"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1474","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":""},"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":"1211.1474","created_at":"2026-05-18T03:26:08.745409+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.1474v2","created_at":"2026-05-18T03:26:08.745409+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1474","created_at":"2026-05-18T03:26:08.745409+00:00"},{"alias_kind":"pith_short_12","alias_value":"3G47YG2P65QE","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_16","alias_value":"3G47YG2P65QEZQHO","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_8","alias_value":"3G47YG2P","created_at":"2026-05-18T12:26:50.516681+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/3G47YG2P65QEZQHOUC2JJWMJBE","json":"https://pith.science/pith/3G47YG2P65QEZQHOUC2JJWMJBE.json","graph_json":"https://pith.science/api/pith-number/3G47YG2P65QEZQHOUC2JJWMJBE/graph.json","events_json":"https://pith.science/api/pith-number/3G47YG2P65QEZQHOUC2JJWMJBE/events.json","paper":"https://pith.science/paper/3G47YG2P"},"agent_actions":{"view_html":"https://pith.science/pith/3G47YG2P65QEZQHOUC2JJWMJBE","download_json":"https://pith.science/pith/3G47YG2P65QEZQHOUC2JJWMJBE.json","view_paper":"https://pith.science/paper/3G47YG2P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.1474&json=true","fetch_graph":"https://pith.science/api/pith-number/3G47YG2P65QEZQHOUC2JJWMJBE/graph.json","fetch_events":"https://pith.science/api/pith-number/3G47YG2P65QEZQHOUC2JJWMJBE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3G47YG2P65QEZQHOUC2JJWMJBE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3G47YG2P65QEZQHOUC2JJWMJBE/action/storage_attestation","attest_author":"https://pith.science/pith/3G47YG2P65QEZQHOUC2JJWMJBE/action/author_attestation","sign_citation":"https://pith.science/pith/3G47YG2P65QEZQHOUC2JJWMJBE/action/citation_signature","submit_replication":"https://pith.science/pith/3G47YG2P65QEZQHOUC2JJWMJBE/action/replication_record"}},"created_at":"2026-05-18T03:26:08.745409+00:00","updated_at":"2026-05-18T03:26:08.745409+00:00"}