{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:5EJXJESHG2QIHHDWCQRFGREKAR","short_pith_number":"pith:5EJXJESH","schema_version":"1.0","canonical_sha256":"e91374924736a0839c76142253448a046af5993e2a6f800803f200460b976322","source":{"kind":"arxiv","id":"1803.11468","version":1},"attestation_state":"computed","paper":{"title":"$L^2$-harmonic $p$-forms on submanifolds with finite total curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jundong Zhou","submitted_at":"2018-03-29T08:15:30Z","abstract_excerpt":"Let $H^p(L^2(M))$ be the space of all $L^2$-harmonic $p$-forms $(2\\leq p\\leq n-2)$ on complete submanifolds $M$ with flat normal bundle in spheres. In this paper, we first show that $H^p(L^2(M))$ is trivial if the total curvature of $M$ is less than a positive constant depending only on $n$. Second, we show that the dimension of $H^p(L^2(M))$ is finite if the total curvature of $M$ is finite. The vanishing theorem is a generalized version of Gan-Zhu-Fang theorem and the finiteness theorem is an extension of Zhu-Fang theorem."},"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":"1803.11468","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-03-29T08:15:30Z","cross_cats_sorted":[],"title_canon_sha256":"a3f8239f1077a5fed8630c38d246d039a3722950d64e779b975254aa22e970b4","abstract_canon_sha256":"b0b94eb3d7139e7cca45df3e72529ed5a9f1efb0cf88db3cd421ce0c3982d937"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:43.493421Z","signature_b64":"cAmDrZDjgGtdKm7FYke6K2ssT9mW4axN51ip8FBAdPcbbVLcDhNwBRa+aNS2nlme30K8Q6arnRkmnuhdB2d3Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e91374924736a0839c76142253448a046af5993e2a6f800803f200460b976322","last_reissued_at":"2026-05-18T00:19:43.492581Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:43.492581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$L^2$-harmonic $p$-forms on submanifolds with finite total curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jundong Zhou","submitted_at":"2018-03-29T08:15:30Z","abstract_excerpt":"Let $H^p(L^2(M))$ be the space of all $L^2$-harmonic $p$-forms $(2\\leq p\\leq n-2)$ on complete submanifolds $M$ with flat normal bundle in spheres. In this paper, we first show that $H^p(L^2(M))$ is trivial if the total curvature of $M$ is less than a positive constant depending only on $n$. Second, we show that the dimension of $H^p(L^2(M))$ is finite if the total curvature of $M$ is finite. The vanishing theorem is a generalized version of Gan-Zhu-Fang theorem and the finiteness theorem is an extension of Zhu-Fang theorem."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.11468","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":"1803.11468","created_at":"2026-05-18T00:19:43.492723+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.11468v1","created_at":"2026-05-18T00:19:43.492723+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.11468","created_at":"2026-05-18T00:19:43.492723+00:00"},{"alias_kind":"pith_short_12","alias_value":"5EJXJESHG2QI","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_16","alias_value":"5EJXJESHG2QIHHDW","created_at":"2026-05-18T12:32:08.215937+00:00"},{"alias_kind":"pith_short_8","alias_value":"5EJXJESH","created_at":"2026-05-18T12:32:08.215937+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/5EJXJESHG2QIHHDWCQRFGREKAR","json":"https://pith.science/pith/5EJXJESHG2QIHHDWCQRFGREKAR.json","graph_json":"https://pith.science/api/pith-number/5EJXJESHG2QIHHDWCQRFGREKAR/graph.json","events_json":"https://pith.science/api/pith-number/5EJXJESHG2QIHHDWCQRFGREKAR/events.json","paper":"https://pith.science/paper/5EJXJESH"},"agent_actions":{"view_html":"https://pith.science/pith/5EJXJESHG2QIHHDWCQRFGREKAR","download_json":"https://pith.science/pith/5EJXJESHG2QIHHDWCQRFGREKAR.json","view_paper":"https://pith.science/paper/5EJXJESH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.11468&json=true","fetch_graph":"https://pith.science/api/pith-number/5EJXJESHG2QIHHDWCQRFGREKAR/graph.json","fetch_events":"https://pith.science/api/pith-number/5EJXJESHG2QIHHDWCQRFGREKAR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5EJXJESHG2QIHHDWCQRFGREKAR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5EJXJESHG2QIHHDWCQRFGREKAR/action/storage_attestation","attest_author":"https://pith.science/pith/5EJXJESHG2QIHHDWCQRFGREKAR/action/author_attestation","sign_citation":"https://pith.science/pith/5EJXJESHG2QIHHDWCQRFGREKAR/action/citation_signature","submit_replication":"https://pith.science/pith/5EJXJESHG2QIHHDWCQRFGREKAR/action/replication_record"}},"created_at":"2026-05-18T00:19:43.492723+00:00","updated_at":"2026-05-18T00:19:43.492723+00:00"}