{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:N6IVMHXY4NTW7LVWICBL6WR3J6","short_pith_number":"pith:N6IVMHXY","schema_version":"1.0","canonical_sha256":"6f91561ef8e3676faeb64082bf5a3b4f9a0836e2dc12a89a6ba28c5b3b5054df","source":{"kind":"arxiv","id":"1802.00226","version":3},"attestation_state":"computed","paper":{"title":"Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Felix Schulze","submitted_at":"2018-02-01T10:20:39Z","abstract_excerpt":"Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $\\Sigma$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp. flat chain mod 2) such that $\\partial S = \\Sigma$. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from $\\Sigma$, to show that S satisfies the optimal Euclidean isoperimetric inequality: $ 6 \\sqrt{\\pi}\\, \\mathbf{M}[S] \\leq (\\mathbf{M}[\\Sigma])^{3/2} $. We also obtain an optimal estimate in case the sectional curvatures "},"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":"1802.00226","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-02-01T10:20:39Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"9b95d7acafe774d78a08a24e89d1dfffe37d4e1f570ce45e5a6ec3cb7ffb3c77","abstract_canon_sha256":"a3ec10a8404a69bdb12cb0e97b111141a7cea216e11a37b9a36d2ad67085a5ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:37:59.763479Z","signature_b64":"h/B/V/FJS+CsXo6hbk9ZdfuWT+tAENDp5y4MsFlQnUpCYOE76Zetq/H32nnTCzZp+G3dxI0QUkQ397PiOwUnAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f91561ef8e3676faeb64082bf5a3b4f9a0836e2dc12a89a6ba28c5b3b5054df","last_reissued_at":"2026-07-05T00:37:59.763010Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:37:59.763010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Felix Schulze","submitted_at":"2018-02-01T10:20:39Z","abstract_excerpt":"Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $\\Sigma$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp. flat chain mod 2) such that $\\partial S = \\Sigma$. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from $\\Sigma$, to show that S satisfies the optimal Euclidean isoperimetric inequality: $ 6 \\sqrt{\\pi}\\, \\mathbf{M}[S] \\leq (\\mathbf{M}[\\Sigma])^{3/2} $. We also obtain an optimal estimate in case the sectional curvatures "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.00226","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1802.00226/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":"1802.00226","created_at":"2026-07-05T00:37:59.763066+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.00226v3","created_at":"2026-07-05T00:37:59.763066+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.00226","created_at":"2026-07-05T00:37:59.763066+00:00"},{"alias_kind":"pith_short_12","alias_value":"N6IVMHXY4NTW","created_at":"2026-07-05T00:37:59.763066+00:00"},{"alias_kind":"pith_short_16","alias_value":"N6IVMHXY4NTW7LVW","created_at":"2026-07-05T00:37:59.763066+00:00"},{"alias_kind":"pith_short_8","alias_value":"N6IVMHXY","created_at":"2026-07-05T00:37:59.763066+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.09814","citing_title":"Total curvature and the isoperimetric inequality in Cartan-Hadamard manifolds","ref_index":132,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6","json":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6.json","graph_json":"https://pith.science/api/pith-number/N6IVMHXY4NTW7LVWICBL6WR3J6/graph.json","events_json":"https://pith.science/api/pith-number/N6IVMHXY4NTW7LVWICBL6WR3J6/events.json","paper":"https://pith.science/paper/N6IVMHXY"},"agent_actions":{"view_html":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6","download_json":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6.json","view_paper":"https://pith.science/paper/N6IVMHXY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.00226&json=true","fetch_graph":"https://pith.science/api/pith-number/N6IVMHXY4NTW7LVWICBL6WR3J6/graph.json","fetch_events":"https://pith.science/api/pith-number/N6IVMHXY4NTW7LVWICBL6WR3J6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6/action/storage_attestation","attest_author":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6/action/author_attestation","sign_citation":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6/action/citation_signature","submit_replication":"https://pith.science/pith/N6IVMHXY4NTW7LVWICBL6WR3J6/action/replication_record"}},"created_at":"2026-07-05T00:37:59.763066+00:00","updated_at":"2026-07-05T00:37:59.763066+00:00"}