{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:FHIDJ5BZB2IP37D5N5GKDVOK7M","short_pith_number":"pith:FHIDJ5BZ","canonical_record":{"source":{"id":"1607.08402","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-28T11:10:01Z","cross_cats_sorted":[],"title_canon_sha256":"48024d6ee00916a8efcb55246d5d0cb92fd35052f7f2fe90210b518efdf042aa","abstract_canon_sha256":"5013b17f6f1d6db7ce8bd97745a54d12af95ada9f20e701df95ade08264c8cfe"},"schema_version":"1.0"},"canonical_sha256":"29d034f4390e90fdfc7d6f4ca1d5cafb19e5afe45bb3e9482bb54e98407d865d","source":{"kind":"arxiv","id":"1607.08402","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.08402","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"arxiv_version","alias_value":"1607.08402v1","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.08402","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"pith_short_12","alias_value":"FHIDJ5BZB2IP","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FHIDJ5BZB2IP37D5","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FHIDJ5BZ","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:FHIDJ5BZB2IP37D5N5GKDVOK7M","target":"record","payload":{"canonical_record":{"source":{"id":"1607.08402","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-28T11:10:01Z","cross_cats_sorted":[],"title_canon_sha256":"48024d6ee00916a8efcb55246d5d0cb92fd35052f7f2fe90210b518efdf042aa","abstract_canon_sha256":"5013b17f6f1d6db7ce8bd97745a54d12af95ada9f20e701df95ade08264c8cfe"},"schema_version":"1.0"},"canonical_sha256":"29d034f4390e90fdfc7d6f4ca1d5cafb19e5afe45bb3e9482bb54e98407d865d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:19.214758Z","signature_b64":"DqdntjBiSq9cY5WJseGSA6CsBt7fAt99i7NfrU1Uegav3vAGFI5jvn8nff6jgiwlV5JaI3ctm6jCP2BrZEAxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29d034f4390e90fdfc7d6f4ca1d5cafb19e5afe45bb3e9482bb54e98407d865d","last_reissued_at":"2026-05-18T01:10:19.214243Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:19.214243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.08402","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":"2026-05-18T01:10:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kku8cgW0ynTLUuj+DbOLlShpmZ9UkPpa0pGTMfFg0ypjvi1uq2CaaRhppeGPaw30UadGsXcLSzpspypsvvUzCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T05:06:36.731262Z"},"content_sha256":"027c6cde78a7dd94521b8cd089303ba0c1171067917863205e2100f4ca357af6","schema_version":"1.0","event_id":"sha256:027c6cde78a7dd94521b8cd089303ba0c1171067917863205e2100f4ca357af6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:FHIDJ5BZB2IP37D5N5GKDVOK7M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Type I singularities in the curve shortening flow associated to a density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Francisco Vi\\~nado-Lereu, Vicente Miquel","submitted_at":"2016-07-28T11:10:01Z","abstract_excerpt":"We define Type I singularities for the mean curvature flow associated to a density $\\psi$ ($\\psi$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $\\psi$-curvature due to the density. We describe a family of curves whose evolution under $\\psi$MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.08402","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":"2026-05-18T01:10:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mwtfxio3WLeQBWMT4GnLXkVC/qqDqIomfR0vvdq8zWAJy1Xs1iZm9dUmdJ2xyVOZVj5WOcmJjXyfJzlTv/eaAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T05:06:36.731737Z"},"content_sha256":"bc737459b643a4e916ed5372866454b0dc599e63b095893c1584432e5e251d1d","schema_version":"1.0","event_id":"sha256:bc737459b643a4e916ed5372866454b0dc599e63b095893c1584432e5e251d1d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FHIDJ5BZB2IP37D5N5GKDVOK7M/bundle.json","state_url":"https://pith.science/pith/FHIDJ5BZB2IP37D5N5GKDVOK7M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FHIDJ5BZB2IP37D5N5GKDVOK7M/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":"2026-05-30T05:06:36Z","links":{"resolver":"https://pith.science/pith/FHIDJ5BZB2IP37D5N5GKDVOK7M","bundle":"https://pith.science/pith/FHIDJ5BZB2IP37D5N5GKDVOK7M/bundle.json","state":"https://pith.science/pith/FHIDJ5BZB2IP37D5N5GKDVOK7M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FHIDJ5BZB2IP37D5N5GKDVOK7M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:FHIDJ5BZB2IP37D5N5GKDVOK7M","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":"5013b17f6f1d6db7ce8bd97745a54d12af95ada9f20e701df95ade08264c8cfe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-28T11:10:01Z","title_canon_sha256":"48024d6ee00916a8efcb55246d5d0cb92fd35052f7f2fe90210b518efdf042aa"},"schema_version":"1.0","source":{"id":"1607.08402","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.08402","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"arxiv_version","alias_value":"1607.08402v1","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.08402","created_at":"2026-05-18T01:10:19Z"},{"alias_kind":"pith_short_12","alias_value":"FHIDJ5BZB2IP","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"FHIDJ5BZB2IP37D5","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"FHIDJ5BZ","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:bc737459b643a4e916ed5372866454b0dc599e63b095893c1584432e5e251d1d","target":"graph","created_at":"2026-05-18T01:10:19Z","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 define Type I singularities for the mean curvature flow associated to a density $\\psi$ ($\\psi$MCF) and describe the blow-up at singular time of these singularities. Special attention is paid to the case where the singularity come from the part of the $\\psi$-curvature due to the density. We describe a family of curves whose evolution under $\\psi$MCF (in a Riemannian surface of non-negative curvature with a density which is singular at a geodesic of the surface) produces only type I singularities and study the limits of their blow-ups.","authors_text":"Francisco Vi\\~nado-Lereu, Vicente Miquel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-28T11:10:01Z","title":"Type I singularities in the curve shortening flow associated to a density"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.08402","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":"sha256:027c6cde78a7dd94521b8cd089303ba0c1171067917863205e2100f4ca357af6","target":"record","created_at":"2026-05-18T01:10:19Z","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":"5013b17f6f1d6db7ce8bd97745a54d12af95ada9f20e701df95ade08264c8cfe","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2016-07-28T11:10:01Z","title_canon_sha256":"48024d6ee00916a8efcb55246d5d0cb92fd35052f7f2fe90210b518efdf042aa"},"schema_version":"1.0","source":{"id":"1607.08402","kind":"arxiv","version":1}},"canonical_sha256":"29d034f4390e90fdfc7d6f4ca1d5cafb19e5afe45bb3e9482bb54e98407d865d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29d034f4390e90fdfc7d6f4ca1d5cafb19e5afe45bb3e9482bb54e98407d865d","first_computed_at":"2026-05-18T01:10:19.214243Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:19.214243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DqdntjBiSq9cY5WJseGSA6CsBt7fAt99i7NfrU1Uegav3vAGFI5jvn8nff6jgiwlV5JaI3ctm6jCP2BrZEAxAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:19.214758Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.08402","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:027c6cde78a7dd94521b8cd089303ba0c1171067917863205e2100f4ca357af6","sha256:bc737459b643a4e916ed5372866454b0dc599e63b095893c1584432e5e251d1d"],"state_sha256":"cdbbfeb8afc967dc7b1eb9969624f46e0f80a164b5ebceee5b64d4bbe57b4799"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BHS2NoFtXzElLvzWlRTK08tty5IAJnqM90HH8FuCctiGdCgpOQ1bcQIJU2Ng13S/QZaVTewmD7ekxAQdPgwcCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T05:06:36.734778Z","bundle_sha256":"cf8dbcb911f2ae58a7849d7d5b15d7cfa493b499cd3fe4e5e332a1427bbbdb32"}}