{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:KR4CH4OYH46CQAY6EVEJFCJQHP","short_pith_number":"pith:KR4CH4OY","canonical_record":{"source":{"id":"1712.03870","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2017-12-11T16:25:30Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c67d4ba9761bc98d106b31ea78611c10bf52347fa622785fff57bafa12f50d90","abstract_canon_sha256":"a228adb203b433b0f25c87caed02044cab52109770ba483b0f3567ef43276667"},"schema_version":"1.0"},"canonical_sha256":"547823f1d83f3c28031e25489289303bdf1b67b57c268b7616944d688eefde92","source":{"kind":"arxiv","id":"1712.03870","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.03870","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"arxiv_version","alias_value":"1712.03870v2","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.03870","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"pith_short_12","alias_value":"KR4CH4OYH46C","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_16","alias_value":"KR4CH4OYH46CQAY6","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_8","alias_value":"KR4CH4OY","created_at":"2026-05-18T12:31:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:KR4CH4OYH46CQAY6EVEJFCJQHP","target":"record","payload":{"canonical_record":{"source":{"id":"1712.03870","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2017-12-11T16:25:30Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c67d4ba9761bc98d106b31ea78611c10bf52347fa622785fff57bafa12f50d90","abstract_canon_sha256":"a228adb203b433b0f25c87caed02044cab52109770ba483b0f3567ef43276667"},"schema_version":"1.0"},"canonical_sha256":"547823f1d83f3c28031e25489289303bdf1b67b57c268b7616944d688eefde92","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:04.135281Z","signature_b64":"28bSApV/b7wiw6hncRcDc9LAgyXEolloQAmJakRGwVMIfo7zLNM5ztIbSQCWDu2OUPf0iVAk58JWHHoXmn0RDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"547823f1d83f3c28031e25489289303bdf1b67b57c268b7616944d688eefde92","last_reissued_at":"2026-05-18T00:09:04.134577Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:04.134577Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.03870","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":"2026-05-18T00:09:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d514RlPv7jQCGq3C+EZwxCfOS+3KY5oCdK3NNX2AHA7eYJm4snfAj2yM2cFxl3DVgv+2CzHKBTwP/3kxJ65gBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:24:17.085192Z"},"content_sha256":"19eb551220add6c7ee6784f827a4fe6ad0c1cd95f99f2f6b97f68e45480a6020","schema_version":"1.0","event_id":"sha256:19eb551220add6c7ee6784f827a4fe6ad0c1cd95f99f2f6b97f68e45480a6020"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:KR4CH4OYH46CQAY6EVEJFCJQHP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Volker Branding","submitted_at":"2017-12-11T16:25:30Z","abstract_excerpt":"We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \\(n\\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the $L^p$-norm of the tension field is bounded and the $n$-energy of the map is sufficiently small then every biharmonic map must be harmonic, where \\(2<p<n\\)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.03870","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":"2026-05-18T00:09:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Kd/0CUE9+5qToPLkcPWjOHnqAaN1sVztC9kFafV+YMrcItfaWUgECz9bqiN9hizhB8hifV3mHLXpHGDIra3sAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T16:24:17.085574Z"},"content_sha256":"91270a7671698159f8018175d4fe18542e2368e8acd2d069fba9cfc025fe706c","schema_version":"1.0","event_id":"sha256:91270a7671698159f8018175d4fe18542e2368e8acd2d069fba9cfc025fe706c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KR4CH4OYH46CQAY6EVEJFCJQHP/bundle.json","state_url":"https://pith.science/pith/KR4CH4OYH46CQAY6EVEJFCJQHP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KR4CH4OYH46CQAY6EVEJFCJQHP/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":"2026-06-23T16:24:17Z","links":{"resolver":"https://pith.science/pith/KR4CH4OYH46CQAY6EVEJFCJQHP","bundle":"https://pith.science/pith/KR4CH4OYH46CQAY6EVEJFCJQHP/bundle.json","state":"https://pith.science/pith/KR4CH4OYH46CQAY6EVEJFCJQHP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KR4CH4OYH46CQAY6EVEJFCJQHP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:KR4CH4OYH46CQAY6EVEJFCJQHP","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":"a228adb203b433b0f25c87caed02044cab52109770ba483b0f3567ef43276667","cross_cats_sorted":["math.AP"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2017-12-11T16:25:30Z","title_canon_sha256":"c67d4ba9761bc98d106b31ea78611c10bf52347fa622785fff57bafa12f50d90"},"schema_version":"1.0","source":{"id":"1712.03870","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.03870","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"arxiv_version","alias_value":"1712.03870v2","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.03870","created_at":"2026-05-18T00:09:04Z"},{"alias_kind":"pith_short_12","alias_value":"KR4CH4OYH46C","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_16","alias_value":"KR4CH4OYH46CQAY6","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_8","alias_value":"KR4CH4OY","created_at":"2026-05-18T12:31:28Z"}],"graph_snapshots":[{"event_id":"sha256:91270a7671698159f8018175d4fe18542e2368e8acd2d069fba9cfc025fe706c","target":"graph","created_at":"2026-05-18T00:09:04Z","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":"We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \\(n\\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the $L^p$-norm of the tension field is bounded and the $n$-energy of the map is sufficiently small then every biharmonic map must be harmonic, where \\(2<p<n\\).","authors_text":"Volker Branding","cross_cats":["math.AP"],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2017-12-11T16:25:30Z","title":"A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.03870","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":"sha256:19eb551220add6c7ee6784f827a4fe6ad0c1cd95f99f2f6b97f68e45480a6020","target":"record","created_at":"2026-05-18T00:09:04Z","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":"a228adb203b433b0f25c87caed02044cab52109770ba483b0f3567ef43276667","cross_cats_sorted":["math.AP"],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.DG","submitted_at":"2017-12-11T16:25:30Z","title_canon_sha256":"c67d4ba9761bc98d106b31ea78611c10bf52347fa622785fff57bafa12f50d90"},"schema_version":"1.0","source":{"id":"1712.03870","kind":"arxiv","version":2}},"canonical_sha256":"547823f1d83f3c28031e25489289303bdf1b67b57c268b7616944d688eefde92","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"547823f1d83f3c28031e25489289303bdf1b67b57c268b7616944d688eefde92","first_computed_at":"2026-05-18T00:09:04.134577Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:04.134577Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"28bSApV/b7wiw6hncRcDc9LAgyXEolloQAmJakRGwVMIfo7zLNM5ztIbSQCWDu2OUPf0iVAk58JWHHoXmn0RDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:04.135281Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.03870","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19eb551220add6c7ee6784f827a4fe6ad0c1cd95f99f2f6b97f68e45480a6020","sha256:91270a7671698159f8018175d4fe18542e2368e8acd2d069fba9cfc025fe706c"],"state_sha256":"52f7f397d495a15ccdcc07e5cae0e831dc17f56def507aac04c25a2fa0e92440"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ruk3ic8cFYdHRqGt6Fav01jjVCRfXEhPAKAytEJj/L1PALWfLOA05qt7EGzzTxAYkfzcXfyoaxlaiswpcgKxDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T16:24:17.087560Z","bundle_sha256":"c6fa68b97e786941eaf636b3077dfdba3c6155f73276d736a095a2786554b90f"}}