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We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\\mathbb{R}$ \\begin{equation}\n  (-\\Delta)^\\frac{1}{2} u =Ke^u \\quad \\text{in }\\mathbb{R}, \\end{equation} with $K$ "},"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":"1503.08701","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2015-03-30T15:08:42Z","cross_cats_sorted":["math.AP","math.CV"],"title_canon_sha256":"3aa38ecd2c4e9dc0c72cc1af89062a6e2d67d92838964fd15884ee138b0bc278","abstract_canon_sha256":"b7a5961544b0f403ce294e89f412c96bb8e06ffe2f6d135db8bad7055d069443"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:21.286215Z","signature_b64":"n+6GpD6Ww3n9yzthfZ+EqNtaMu7tPORuTTlkddNXztvN/GxR1YUlyshBamRT8L4M06jApWZs6+2Z674VJFT4Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"023ea6b7489ac21b8189952110cb0ad2822b85d02e714970eb557aa5fc382c66","last_reissued_at":"2026-05-18T01:22:21.285451Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:21.285451Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Blow-up analysis of a nonlocal Liouville-type equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CV"],"primary_cat":"math.DG","authors_text":"Francesca Da Lio, Luca Martinazzi, Tristan Rivi\\`ere","submitted_at":"2015-03-30T15:08:42Z","abstract_excerpt":"In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \\begin{equation}(-\\Delta)^\\frac12 u= \\kappa e^u-1~\\mbox{in $S^1$,} \\end{equation} where $(-\\Delta)^\\frac{1}{2}$ stands for the fractional Laplacian and $\\kappa$ is a bounded function. We interpret the above equation as the prescribed curvature equation to a curve in conformal parametrization. We also establish a relation between this equation and the analogous equation in $\\mathbb{R}$ \\begin{equation}\n  (-\\Delta)^\\frac{1}{2} u =Ke^u \\quad \\text{in }\\mathbb{R}, \\end{equation} with $K$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08701","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":"1503.08701","created_at":"2026-05-18T01:22:21.285564+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.08701v1","created_at":"2026-05-18T01:22:21.285564+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08701","created_at":"2026-05-18T01:22:21.285564+00:00"},{"alias_kind":"pith_short_12","alias_value":"AI7KNN2ITLBB","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"AI7KNN2ITLBBXAMJ","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"AI7KNN2I","created_at":"2026-05-18T12:29:10.953037+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/AI7KNN2ITLBBXAMJSUQRBSYK2K","json":"https://pith.science/pith/AI7KNN2ITLBBXAMJSUQRBSYK2K.json","graph_json":"https://pith.science/api/pith-number/AI7KNN2ITLBBXAMJSUQRBSYK2K/graph.json","events_json":"https://pith.science/api/pith-number/AI7KNN2ITLBBXAMJSUQRBSYK2K/events.json","paper":"https://pith.science/paper/AI7KNN2I"},"agent_actions":{"view_html":"https://pith.science/pith/AI7KNN2ITLBBXAMJSUQRBSYK2K","download_json":"https://pith.science/pith/AI7KNN2ITLBBXAMJSUQRBSYK2K.json","view_paper":"https://pith.science/paper/AI7KNN2I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.08701&json=true","fetch_graph":"https://pith.science/api/pith-number/AI7KNN2ITLBBXAMJSUQRBSYK2K/graph.json","fetch_events":"https://pith.science/api/pith-number/AI7KNN2ITLBBXAMJSUQRBSYK2K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AI7KNN2ITLBBXAMJSUQRBSYK2K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AI7KNN2ITLBBXAMJSUQRBSYK2K/action/storage_attestation","attest_author":"https://pith.science/pith/AI7KNN2ITLBBXAMJSUQRBSYK2K/action/author_attestation","sign_citation":"https://pith.science/pith/AI7KNN2ITLBBXAMJSUQRBSYK2K/action/citation_signature","submit_replication":"https://pith.science/pith/AI7KNN2ITLBBXAMJSUQRBSYK2K/action/replication_record"}},"created_at":"2026-05-18T01:22:21.285564+00:00","updated_at":"2026-05-18T01:22:21.285564+00:00"}