{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:ZGWDFO2NBMKZ7HZLMXC6XYJOXS","short_pith_number":"pith:ZGWDFO2N","schema_version":"1.0","canonical_sha256":"c9ac32bb4d0b159f9f2b65c5ebe12ebc83deeeea73ff6eb564c77948e917c41f","source":{"kind":"arxiv","id":"1307.1876","version":1},"attestation_state":"computed","paper":{"title":"Submanifolds of warped product manifolds $I\\times_f S^{m-1}(k)$}} from a $p$-harmonic viewpoint","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bang-Yen Chen, Shihshu Walter Wei","submitted_at":"2013-07-07T15:22:14Z","abstract_excerpt":"We study $p$-harmonic maps, $p$-harmonic morphisms, biharmonic maps, and quasiregular mappings into submanifolds of warped product Riemannian manifolds ${I}\\times_f S^{m-1}(k)\\, $ of an open interval and a complete simply-connecteded $(m-1)$-dimensional Riemannian manifold of constant sectional curvature $k$. We establish an existence theorem for $p$-harmonic maps and give a classification of complete stable minimal surfaces in certain three dimensional warped product Riemannian manifolds ${\\bf R}\\times_f S^{2}(k)\\, ,$ building on our previous work. When $f \\equiv\\, $ Const. and $k=0$, we reca"},"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":"1307.1876","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-07T15:22:14Z","cross_cats_sorted":[],"title_canon_sha256":"c1c62f6333d77331bb666b2f9773e73100be10127a8db1958c4e38c533209115","abstract_canon_sha256":"41d77186ba4890e9c7d2792e40f63509a40f2f12c54250a161db99aa57fae715"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:03.277295Z","signature_b64":"d6YDV5f24n425RkPANOjBCVdPICOzS93b4qlxAltHFzpTBcvVn3/zY7hq+KbmmBXEHOvGEtjDUMjU6F3tFKABg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9ac32bb4d0b159f9f2b65c5ebe12ebc83deeeea73ff6eb564c77948e917c41f","last_reissued_at":"2026-05-18T03:19:03.276854Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:03.276854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Submanifolds of warped product manifolds $I\\times_f S^{m-1}(k)$}} from a $p$-harmonic viewpoint","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bang-Yen Chen, Shihshu Walter Wei","submitted_at":"2013-07-07T15:22:14Z","abstract_excerpt":"We study $p$-harmonic maps, $p$-harmonic morphisms, biharmonic maps, and quasiregular mappings into submanifolds of warped product Riemannian manifolds ${I}\\times_f S^{m-1}(k)\\, $ of an open interval and a complete simply-connecteded $(m-1)$-dimensional Riemannian manifold of constant sectional curvature $k$. We establish an existence theorem for $p$-harmonic maps and give a classification of complete stable minimal surfaces in certain three dimensional warped product Riemannian manifolds ${\\bf R}\\times_f S^{2}(k)\\, ,$ building on our previous work. When $f \\equiv\\, $ Const. and $k=0$, we reca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.1876","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":"1307.1876","created_at":"2026-05-18T03:19:03.276916+00:00"},{"alias_kind":"arxiv_version","alias_value":"1307.1876v1","created_at":"2026-05-18T03:19:03.276916+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.1876","created_at":"2026-05-18T03:19:03.276916+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZGWDFO2NBMKZ","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZGWDFO2NBMKZ7HZL","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZGWDFO2N","created_at":"2026-05-18T12:28:09.283467+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/ZGWDFO2NBMKZ7HZLMXC6XYJOXS","json":"https://pith.science/pith/ZGWDFO2NBMKZ7HZLMXC6XYJOXS.json","graph_json":"https://pith.science/api/pith-number/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/graph.json","events_json":"https://pith.science/api/pith-number/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/events.json","paper":"https://pith.science/paper/ZGWDFO2N"},"agent_actions":{"view_html":"https://pith.science/pith/ZGWDFO2NBMKZ7HZLMXC6XYJOXS","download_json":"https://pith.science/pith/ZGWDFO2NBMKZ7HZLMXC6XYJOXS.json","view_paper":"https://pith.science/paper/ZGWDFO2N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1307.1876&json=true","fetch_graph":"https://pith.science/api/pith-number/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/graph.json","fetch_events":"https://pith.science/api/pith-number/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/action/storage_attestation","attest_author":"https://pith.science/pith/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/action/author_attestation","sign_citation":"https://pith.science/pith/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/action/citation_signature","submit_replication":"https://pith.science/pith/ZGWDFO2NBMKZ7HZLMXC6XYJOXS/action/replication_record"}},"created_at":"2026-05-18T03:19:03.276916+00:00","updated_at":"2026-05-18T03:19:03.276916+00:00"}