{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:4MRIF4YVKJHWVYI3JJT6DPWG7A","short_pith_number":"pith:4MRIF4YV","schema_version":"1.0","canonical_sha256":"e32282f315524f6ae11b4a67e1bec6f8089a7d7f01a558674ed693769586fa30","source":{"kind":"arxiv","id":"1807.00140","version":1},"attestation_state":"computed","paper":{"title":"A relative entropy for expanders of the Harmonic map flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alix Deruelle","submitted_at":"2018-06-30T08:33:06Z","abstract_excerpt":"In this paper we focus on the uniqueness question for (expanding) solutions of the Harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On the one hand, we prove the existence of two expanding solutions associated to any suitable solution coming out of a $0$-homogeneous map by a blow-up and a blow-down process. On the other hand, generic uniqueness of expanding solutions coming out of the same 0-homogeneous map of 0 relative entropy is proved."},"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":"1807.00140","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2018-06-30T08:33:06Z","cross_cats_sorted":[],"title_canon_sha256":"bc66e02c59344557a64df7acd10ba921307d44e7116c3050be1936a7a4e04da5","abstract_canon_sha256":"bca1707dd0149d083e760d3886ac6220ead2cd720ff1e038bf9c11bd9c6c6ddd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:55.949080Z","signature_b64":"7OAS5izyeYJaRfD3LniZ8En5N1cPhd75YF6LF4QJ/Q7iXEbJ3oYwtX1S9Z7y1U5/6Y1sAkowRIp5VHL3axJLBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e32282f315524f6ae11b4a67e1bec6f8089a7d7f01a558674ed693769586fa30","last_reissued_at":"2026-05-18T00:11:55.948418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:55.948418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A relative entropy for expanders of the Harmonic map flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alix Deruelle","submitted_at":"2018-06-30T08:33:06Z","abstract_excerpt":"In this paper we focus on the uniqueness question for (expanding) solutions of the Harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On the one hand, we prove the existence of two expanding solutions associated to any suitable solution coming out of a $0$-homogeneous map by a blow-up and a blow-down process. On the other hand, generic uniqueness of expanding solutions coming out of the same 0-homogeneous map of 0 relative entropy is proved."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.00140","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":"1807.00140","created_at":"2026-05-18T00:11:55.948515+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.00140v1","created_at":"2026-05-18T00:11:55.948515+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.00140","created_at":"2026-05-18T00:11:55.948515+00:00"},{"alias_kind":"pith_short_12","alias_value":"4MRIF4YVKJHW","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"4MRIF4YVKJHWVYI3","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"4MRIF4YV","created_at":"2026-05-18T12:32:05.422762+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/4MRIF4YVKJHWVYI3JJT6DPWG7A","json":"https://pith.science/pith/4MRIF4YVKJHWVYI3JJT6DPWG7A.json","graph_json":"https://pith.science/api/pith-number/4MRIF4YVKJHWVYI3JJT6DPWG7A/graph.json","events_json":"https://pith.science/api/pith-number/4MRIF4YVKJHWVYI3JJT6DPWG7A/events.json","paper":"https://pith.science/paper/4MRIF4YV"},"agent_actions":{"view_html":"https://pith.science/pith/4MRIF4YVKJHWVYI3JJT6DPWG7A","download_json":"https://pith.science/pith/4MRIF4YVKJHWVYI3JJT6DPWG7A.json","view_paper":"https://pith.science/paper/4MRIF4YV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.00140&json=true","fetch_graph":"https://pith.science/api/pith-number/4MRIF4YVKJHWVYI3JJT6DPWG7A/graph.json","fetch_events":"https://pith.science/api/pith-number/4MRIF4YVKJHWVYI3JJT6DPWG7A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4MRIF4YVKJHWVYI3JJT6DPWG7A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4MRIF4YVKJHWVYI3JJT6DPWG7A/action/storage_attestation","attest_author":"https://pith.science/pith/4MRIF4YVKJHWVYI3JJT6DPWG7A/action/author_attestation","sign_citation":"https://pith.science/pith/4MRIF4YVKJHWVYI3JJT6DPWG7A/action/citation_signature","submit_replication":"https://pith.science/pith/4MRIF4YVKJHWVYI3JJT6DPWG7A/action/replication_record"}},"created_at":"2026-05-18T00:11:55.948515+00:00","updated_at":"2026-05-18T00:11:55.948515+00:00"}