{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:7JZTO4A5VZNKJI66ETU3LHFMOA","short_pith_number":"pith:7JZTO4A5","canonical_record":{"source":{"id":"1705.07976","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-05-22T20:03:40Z","cross_cats_sorted":[],"title_canon_sha256":"28bfaa62d72695e9e705d83f923c33ec4cb9fff82b6703cad09cb51ef9fbfe20","abstract_canon_sha256":"ea6a4cd62b8131f934baad163a57fc406992b1c478011edb24a12bec62d57280"},"schema_version":"1.0"},"canonical_sha256":"fa7337701dae5aa4a3de24e9b59cac7028002e1e387ac8c939b308dc8c16982d","source":{"kind":"arxiv","id":"1705.07976","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.07976","created_at":"2026-05-18T00:44:04Z"},{"alias_kind":"arxiv_version","alias_value":"1705.07976v1","created_at":"2026-05-18T00:44:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.07976","created_at":"2026-05-18T00:44:04Z"},{"alias_kind":"pith_short_12","alias_value":"7JZTO4A5VZNK","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7JZTO4A5VZNKJI66","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7JZTO4A5","created_at":"2026-05-18T12:31:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:7JZTO4A5VZNKJI66ETU3LHFMOA","target":"record","payload":{"canonical_record":{"source":{"id":"1705.07976","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-05-22T20:03:40Z","cross_cats_sorted":[],"title_canon_sha256":"28bfaa62d72695e9e705d83f923c33ec4cb9fff82b6703cad09cb51ef9fbfe20","abstract_canon_sha256":"ea6a4cd62b8131f934baad163a57fc406992b1c478011edb24a12bec62d57280"},"schema_version":"1.0"},"canonical_sha256":"fa7337701dae5aa4a3de24e9b59cac7028002e1e387ac8c939b308dc8c16982d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:04.002112Z","signature_b64":"zIMWdgBNfAeH3hWR67pImNQZN2RZVTCbSgBzJ9ZHHvBcs3t5Kdh+nJFhE8HGaY5x/lNA+rLKgcQ3xGMX+BHYCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa7337701dae5aa4a3de24e9b59cac7028002e1e387ac8c939b308dc8c16982d","last_reissued_at":"2026-05-18T00:44:04.001644Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:04.001644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.07976","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:44:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4GRBGUSbhOMSKRdhYONDWDhG2noJK9DxdkUu9YhHnZcwMemcu1BCodFiwbG5ZeaTLBPpEd8wHvioxVS00mvBAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T21:35:33.447992Z"},"content_sha256":"3ebd506ab42bab55a3b0ef19000b30fae4b28107f36830a0ecdb71bfa467049e","schema_version":"1.0","event_id":"sha256:3ebd506ab42bab55a3b0ef19000b30fae4b28107f36830a0ecdb71bfa467049e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:7JZTO4A5VZNKJI66ETU3LHFMOA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Completeness of Length-Weighted Sobolev Metrics on the Space of Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jakob M{\\o}ller-Andersen, Martins Bruveris","submitted_at":"2017-05-22T20:03:40Z","abstract_excerpt":"In this article we prove completeness results for Sobolev metrics with nonconstant coefficients on the space of immersed curves and on the space of unparametrized curves. We provide necessary as well as sufficient conditions for the coefficients of the Riemannian metric for the metric to be metrically complete and we construct examples of incomplete metrics. This work is an extension of previous work on completeness of Sobolev metrics with constant coefficients."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.07976","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:44:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VnSBQYdULPBj9yVFBQ/AqD23LO3UeL3MICR1FtFURj0+6/IEOVikjWDkeTyOrGXWld5aasRJGB7AvO8tB211Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T21:35:33.448715Z"},"content_sha256":"a5272055e8b9296dbb2a5cfb65048c06789f6dbe108ebdaed2d454c7222aae11","schema_version":"1.0","event_id":"sha256:a5272055e8b9296dbb2a5cfb65048c06789f6dbe108ebdaed2d454c7222aae11"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7JZTO4A5VZNKJI66ETU3LHFMOA/bundle.json","state_url":"https://pith.science/pith/7JZTO4A5VZNKJI66ETU3LHFMOA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7JZTO4A5VZNKJI66ETU3LHFMOA/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":"2026-06-08T21:35:33Z","links":{"resolver":"https://pith.science/pith/7JZTO4A5VZNKJI66ETU3LHFMOA","bundle":"https://pith.science/pith/7JZTO4A5VZNKJI66ETU3LHFMOA/bundle.json","state":"https://pith.science/pith/7JZTO4A5VZNKJI66ETU3LHFMOA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7JZTO4A5VZNKJI66ETU3LHFMOA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:7JZTO4A5VZNKJI66ETU3LHFMOA","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":"ea6a4cd62b8131f934baad163a57fc406992b1c478011edb24a12bec62d57280","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-05-22T20:03:40Z","title_canon_sha256":"28bfaa62d72695e9e705d83f923c33ec4cb9fff82b6703cad09cb51ef9fbfe20"},"schema_version":"1.0","source":{"id":"1705.07976","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.07976","created_at":"2026-05-18T00:44:04Z"},{"alias_kind":"arxiv_version","alias_value":"1705.07976v1","created_at":"2026-05-18T00:44:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.07976","created_at":"2026-05-18T00:44:04Z"},{"alias_kind":"pith_short_12","alias_value":"7JZTO4A5VZNK","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_16","alias_value":"7JZTO4A5VZNKJI66","created_at":"2026-05-18T12:31:05Z"},{"alias_kind":"pith_short_8","alias_value":"7JZTO4A5","created_at":"2026-05-18T12:31:05Z"}],"graph_snapshots":[{"event_id":"sha256:a5272055e8b9296dbb2a5cfb65048c06789f6dbe108ebdaed2d454c7222aae11","target":"graph","created_at":"2026-05-18T00:44:04Z","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":"In this article we prove completeness results for Sobolev metrics with nonconstant coefficients on the space of immersed curves and on the space of unparametrized curves. We provide necessary as well as sufficient conditions for the coefficients of the Riemannian metric for the metric to be metrically complete and we construct examples of incomplete metrics. This work is an extension of previous work on completeness of Sobolev metrics with constant coefficients.","authors_text":"Jakob M{\\o}ller-Andersen, Martins Bruveris","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-05-22T20:03:40Z","title":"Completeness of Length-Weighted Sobolev Metrics on the Space of Curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.07976","kind":"arxiv","version":1},"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":"sha256:3ebd506ab42bab55a3b0ef19000b30fae4b28107f36830a0ecdb71bfa467049e","target":"record","created_at":"2026-05-18T00:44:04Z","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":"ea6a4cd62b8131f934baad163a57fc406992b1c478011edb24a12bec62d57280","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-05-22T20:03:40Z","title_canon_sha256":"28bfaa62d72695e9e705d83f923c33ec4cb9fff82b6703cad09cb51ef9fbfe20"},"schema_version":"1.0","source":{"id":"1705.07976","kind":"arxiv","version":1}},"canonical_sha256":"fa7337701dae5aa4a3de24e9b59cac7028002e1e387ac8c939b308dc8c16982d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa7337701dae5aa4a3de24e9b59cac7028002e1e387ac8c939b308dc8c16982d","first_computed_at":"2026-05-18T00:44:04.001644Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:04.001644Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zIMWdgBNfAeH3hWR67pImNQZN2RZVTCbSgBzJ9ZHHvBcs3t5Kdh+nJFhE8HGaY5x/lNA+rLKgcQ3xGMX+BHYCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:04.002112Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.07976","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ebd506ab42bab55a3b0ef19000b30fae4b28107f36830a0ecdb71bfa467049e","sha256:a5272055e8b9296dbb2a5cfb65048c06789f6dbe108ebdaed2d454c7222aae11"],"state_sha256":"8d3853bbc291e81120b848fe93a9b5f60495e7e553ffac66e2e86bdf8ef1ca97"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RxJ9HsV7Cg324N1iK0ymQUlQemR+xKCmiVCq8xNmSoDxnInuFJeS1Qi2fZGpWOPip3aPV8ivOlF49b7XyrMkCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T21:35:33.452652Z","bundle_sha256":"88fb6c136d0ffe3c98a77de84348a16454fb629a1e54ef9d18686726ec259249"}}