{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:ZA55JA2SN7FUE7DISFII4GABZU","short_pith_number":"pith:ZA55JA2S","canonical_record":{"source":{"id":"1511.07231","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T14:17:11Z","cross_cats_sorted":[],"title_canon_sha256":"6b58281d06b2eaa7850d08fd1f9f7129e522a5e13355d3ebb563bb782c2da8b4","abstract_canon_sha256":"0b33ee6d829695bb0491cf1791c2d02b73ba1fe5320c734a2b397995b1a64da1"},"schema_version":"1.0"},"canonical_sha256":"c83bd483526fcb427c6891508e1801cd2f8165c160541d3a98968d0445455536","source":{"kind":"arxiv","id":"1511.07231","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.07231","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1511.07231v2","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.07231","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"ZA55JA2SN7FU","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZA55JA2SN7FUE7DI","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZA55JA2S","created_at":"2026-05-18T12:29:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:ZA55JA2SN7FUE7DISFII4GABZU","target":"record","payload":{"canonical_record":{"source":{"id":"1511.07231","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T14:17:11Z","cross_cats_sorted":[],"title_canon_sha256":"6b58281d06b2eaa7850d08fd1f9f7129e522a5e13355d3ebb563bb782c2da8b4","abstract_canon_sha256":"0b33ee6d829695bb0491cf1791c2d02b73ba1fe5320c734a2b397995b1a64da1"},"schema_version":"1.0"},"canonical_sha256":"c83bd483526fcb427c6891508e1801cd2f8165c160541d3a98968d0445455536","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:52.206981Z","signature_b64":"LW5/h34DZWe+y5ConJeOlq4KG7PvGn9OZDc6Fv09ULpvNMzNTKY6jj1eQLsF9aLXpUdkRySuRcyWYjZkj5KJBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c83bd483526fcb427c6891508e1801cd2f8165c160541d3a98968d0445455536","last_reissued_at":"2026-05-18T01:17:52.206233Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:52.206233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.07231","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l8cLG7HC92NoRZoG/mYVcPtFFxKMkPhjs8cNPITDQawTQRzUDzrDJGcCM1k9Ub5T6ltmp8B6xCJnJXfwPeDkAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:34:14.236967Z"},"content_sha256":"fc210edc09c3bd72a28e1eaf2f1a5e73c5611b714633dcaa4f49e1d2330542f0","schema_version":"1.0","event_id":"sha256:fc210edc09c3bd72a28e1eaf2f1a5e73c5611b714633dcaa4f49e1d2330542f0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:ZA55JA2SN7FUE7DISFII4GABZU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Remarks on the nonexistence of biharmonic maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Yong Luo","submitted_at":"2015-11-23T14:17:11Z","abstract_excerpt":"In this short note we study nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $\\phi:(M,g)\\to (N, h)$ is a biharmonic map, where $(M, g)$ is a complete Riemannian manifold and $(N,h)$ a Riemannian manifold with nonpositive sectional curvature, we will prove that $\\phi$ is a harmonic map if one of the following conditions holds: (i) $|d\\phi|$ is bounded in $L^q(M)$ and $ \\int_M|\\tau(\\phi)|^pdv_g<\\infty, $ for some $1\\leq q\\leq\\infty$, $1< p<\\infty$; or (ii) $Vol(M)=\\infty$ and $ \\int_M|\\tau(\\phi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07231","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j9lDiL71x5gU4Xg6E3Q4rBNFmAS7U5nNAOqvUVYOyZ2YAzalqZKd0CoVTjW6pps+g2w+GvNspKCwe6sEUUmNBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:34:14.237339Z"},"content_sha256":"2566615f0a5942823125bfe063ee13c370f0fd3382fce34615c6a81b8b00fb42","schema_version":"1.0","event_id":"sha256:2566615f0a5942823125bfe063ee13c370f0fd3382fce34615c6a81b8b00fb42"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZA55JA2SN7FUE7DISFII4GABZU/bundle.json","state_url":"https://pith.science/pith/ZA55JA2SN7FUE7DISFII4GABZU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZA55JA2SN7FUE7DISFII4GABZU/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-28T08:34:14Z","links":{"resolver":"https://pith.science/pith/ZA55JA2SN7FUE7DISFII4GABZU","bundle":"https://pith.science/pith/ZA55JA2SN7FUE7DISFII4GABZU/bundle.json","state":"https://pith.science/pith/ZA55JA2SN7FUE7DISFII4GABZU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZA55JA2SN7FUE7DISFII4GABZU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:ZA55JA2SN7FUE7DISFII4GABZU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0b33ee6d829695bb0491cf1791c2d02b73ba1fe5320c734a2b397995b1a64da1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T14:17:11Z","title_canon_sha256":"6b58281d06b2eaa7850d08fd1f9f7129e522a5e13355d3ebb563bb782c2da8b4"},"schema_version":"1.0","source":{"id":"1511.07231","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.07231","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1511.07231v2","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.07231","created_at":"2026-05-18T01:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"ZA55JA2SN7FU","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_16","alias_value":"ZA55JA2SN7FUE7DI","created_at":"2026-05-18T12:29:52Z"},{"alias_kind":"pith_short_8","alias_value":"ZA55JA2S","created_at":"2026-05-18T12:29:52Z"}],"graph_snapshots":[{"event_id":"sha256:2566615f0a5942823125bfe063ee13c370f0fd3382fce34615c6a81b8b00fb42","target":"graph","created_at":"2026-05-18T01:17:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this short note we study nonexistence result of biharmonic maps from a complete Riemannian manifold into a Riemannian manifold with nonpositive sectional curvature. Assume that $\\phi:(M,g)\\to (N, h)$ is a biharmonic map, where $(M, g)$ is a complete Riemannian manifold and $(N,h)$ a Riemannian manifold with nonpositive sectional curvature, we will prove that $\\phi$ is a harmonic map if one of the following conditions holds: (i) $|d\\phi|$ is bounded in $L^q(M)$ and $ \\int_M|\\tau(\\phi)|^pdv_g<\\infty, $ for some $1\\leq q\\leq\\infty$, $1< p<\\infty$; or (ii) $Vol(M)=\\infty$ and $ \\int_M|\\tau(\\phi","authors_text":"Yong Luo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T14:17:11Z","title":"Remarks on the nonexistence of biharmonic maps"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07231","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:fc210edc09c3bd72a28e1eaf2f1a5e73c5611b714633dcaa4f49e1d2330542f0","target":"record","created_at":"2026-05-18T01:17:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0b33ee6d829695bb0491cf1791c2d02b73ba1fe5320c734a2b397995b1a64da1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-11-23T14:17:11Z","title_canon_sha256":"6b58281d06b2eaa7850d08fd1f9f7129e522a5e13355d3ebb563bb782c2da8b4"},"schema_version":"1.0","source":{"id":"1511.07231","kind":"arxiv","version":2}},"canonical_sha256":"c83bd483526fcb427c6891508e1801cd2f8165c160541d3a98968d0445455536","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c83bd483526fcb427c6891508e1801cd2f8165c160541d3a98968d0445455536","first_computed_at":"2026-05-18T01:17:52.206233Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:17:52.206233Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LW5/h34DZWe+y5ConJeOlq4KG7PvGn9OZDc6Fv09ULpvNMzNTKY6jj1eQLsF9aLXpUdkRySuRcyWYjZkj5KJBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:17:52.206981Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.07231","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fc210edc09c3bd72a28e1eaf2f1a5e73c5611b714633dcaa4f49e1d2330542f0","sha256:2566615f0a5942823125bfe063ee13c370f0fd3382fce34615c6a81b8b00fb42"],"state_sha256":"624346f00295726aae2c38a83eeeed1081a4c28ed26b962b404b0bc5668e0231"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NPpmpeY2fMvgla7TBRDApGmVIXsibQjbHom6eBjtgU8SzbkwGCRwQKpO90G+5TNTQ4qPOnido4Eo0ukeXMA7CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T08:34:14.240052Z","bundle_sha256":"23d47f5bf3210fbe2e7111e07c42c0b53fe0d97bb479204c11f4d9b1d8bc17e7"}}