{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:RFKWP5TAAINX4NQNXXQ5YKFFW2","short_pith_number":"pith:RFKWP5TA","schema_version":"1.0","canonical_sha256":"895567f660021b7e360dbde1dc28a5b6ab3073249ada7ed099b9724285381dfa","source":{"kind":"arxiv","id":"1902.03161","version":1},"attestation_state":"computed","paper":{"title":"On the size of the singular set of minimizing harmonic maps into the 2-sphere in dimension four and higher","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Armin Schikorra, Katarzyna Mazowiecka, Micha{\\l} Mi\\'skiewicz","submitted_at":"2019-02-08T16:04:10Z","abstract_excerpt":"We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \\geq 4$. For minimizing harmonic maps $u\\in W^{1,2}(\\Omega,\\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove:\n  (1) An extension of Almgren and Lieb's linear law, namely \\[\\mathcal{H}^{n-3}(\\textrm{sing} u) \\le C \\int_{\\partial \\Omega} |\\nabla_T u|^{n-1} \\,d\\mathcal{H}^{n-1};\\]\n  (2) An extension of Hardt and Lin's stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary."},"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":"1902.03161","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-02-08T16:04:10Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"5458a7511b32cec010af9a42aa06e686b6e3d6199fd6d21173554bc6a3451cd1","abstract_canon_sha256":"8fa8b9723a53379cb830a34ae82a25dd68c3ab980f89de21a9eaaadb1f5efea9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:27.773985Z","signature_b64":"Qw48BTbixoTue91cHOZ0QewaBEkkwJ4Mb+Lkk5rPDJlcFbK7ndyEfmynWVgOhYsLyAfOdEo3IyZ1UeB4hp6WAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"895567f660021b7e360dbde1dc28a5b6ab3073249ada7ed099b9724285381dfa","last_reissued_at":"2026-05-17T23:54:27.773297Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:27.773297Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the size of the singular set of minimizing harmonic maps into the 2-sphere in dimension four and higher","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Armin Schikorra, Katarzyna Mazowiecka, Micha{\\l} Mi\\'skiewicz","submitted_at":"2019-02-08T16:04:10Z","abstract_excerpt":"We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \\geq 4$. For minimizing harmonic maps $u\\in W^{1,2}(\\Omega,\\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove:\n  (1) An extension of Almgren and Lieb's linear law, namely \\[\\mathcal{H}^{n-3}(\\textrm{sing} u) \\le C \\int_{\\partial \\Omega} |\\nabla_T u|^{n-1} \\,d\\mathcal{H}^{n-1};\\]\n  (2) An extension of Hardt and Lin's stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.03161","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":""},"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":"1902.03161","created_at":"2026-05-17T23:54:27.773424+00:00"},{"alias_kind":"arxiv_version","alias_value":"1902.03161v1","created_at":"2026-05-17T23:54:27.773424+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.03161","created_at":"2026-05-17T23:54:27.773424+00:00"},{"alias_kind":"pith_short_12","alias_value":"RFKWP5TAAINX","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_16","alias_value":"RFKWP5TAAINX4NQN","created_at":"2026-05-18T12:33:27.125529+00:00"},{"alias_kind":"pith_short_8","alias_value":"RFKWP5TA","created_at":"2026-05-18T12:33:27.125529+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/RFKWP5TAAINX4NQNXXQ5YKFFW2","json":"https://pith.science/pith/RFKWP5TAAINX4NQNXXQ5YKFFW2.json","graph_json":"https://pith.science/api/pith-number/RFKWP5TAAINX4NQNXXQ5YKFFW2/graph.json","events_json":"https://pith.science/api/pith-number/RFKWP5TAAINX4NQNXXQ5YKFFW2/events.json","paper":"https://pith.science/paper/RFKWP5TA"},"agent_actions":{"view_html":"https://pith.science/pith/RFKWP5TAAINX4NQNXXQ5YKFFW2","download_json":"https://pith.science/pith/RFKWP5TAAINX4NQNXXQ5YKFFW2.json","view_paper":"https://pith.science/paper/RFKWP5TA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1902.03161&json=true","fetch_graph":"https://pith.science/api/pith-number/RFKWP5TAAINX4NQNXXQ5YKFFW2/graph.json","fetch_events":"https://pith.science/api/pith-number/RFKWP5TAAINX4NQNXXQ5YKFFW2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RFKWP5TAAINX4NQNXXQ5YKFFW2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RFKWP5TAAINX4NQNXXQ5YKFFW2/action/storage_attestation","attest_author":"https://pith.science/pith/RFKWP5TAAINX4NQNXXQ5YKFFW2/action/author_attestation","sign_citation":"https://pith.science/pith/RFKWP5TAAINX4NQNXXQ5YKFFW2/action/citation_signature","submit_replication":"https://pith.science/pith/RFKWP5TAAINX4NQNXXQ5YKFFW2/action/replication_record"}},"created_at":"2026-05-17T23:54:27.773424+00:00","updated_at":"2026-05-17T23:54:27.773424+00:00"}