{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:IB27PIQJG5GFKFVH4VWXE6PLRA","short_pith_number":"pith:IB27PIQJ","schema_version":"1.0","canonical_sha256":"4075f7a209374c5516a7e56d7279eb88234fe7564b6d2fbeff734c781a433994","source":{"kind":"arxiv","id":"1608.01905","version":1},"attestation_state":"computed","paper":{"title":"Conformally Euclidean metrics on $\\mathbb{R}^n$ with arbitrary total $Q$-curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ali Hyder","submitted_at":"2016-08-05T15:05:42Z","abstract_excerpt":"We study the existence of solution to the problem $$(-\\Delta)^\\frac n2u=Qe^{nu}\\quad\\text{in }\\mathbb{R}^{n},\\quad \\kappa:=\\int_{\\mathbb{R}^{n}}Qe^{nu}dx<\\infty,$$ where $Q\\geq 0$, $\\kappa\\in (0,\\infty)$ and $n\\geq 3$. Using ODE techniques Martinazzi for $n=6$ and Huang-Ye for $n=4m+2$ proved the existence of solution to the above problem with $Q\\equiv const>0$ and for every $\\kappa\\in (0,\\infty)$. We extend these results in every dimension $n\\geq 5$, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which $Q$ is non-constant, and under some d"},"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":"1608.01905","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-08-05T15:05:42Z","cross_cats_sorted":[],"title_canon_sha256":"cd372689e0f45997d48d86a07345d4d8a0af579cf9defc0f0173f21fd5ee5edf","abstract_canon_sha256":"d7d6973d9068a6b9896bab6a18da85a8f577896d9d74e78d76333b2b5b96b7fd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:32.617284Z","signature_b64":"2haov9TBM8CyWgc6+qdx44ltsFrPpiAWhvq21V0whnkur7cxhXqiVg18omc2chs0fzLHWphuN6nCbh4thG23CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4075f7a209374c5516a7e56d7279eb88234fe7564b6d2fbeff734c781a433994","last_reissued_at":"2026-05-18T00:42:32.616642Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:32.616642Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Conformally Euclidean metrics on $\\mathbb{R}^n$ with arbitrary total $Q$-curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Ali Hyder","submitted_at":"2016-08-05T15:05:42Z","abstract_excerpt":"We study the existence of solution to the problem $$(-\\Delta)^\\frac n2u=Qe^{nu}\\quad\\text{in }\\mathbb{R}^{n},\\quad \\kappa:=\\int_{\\mathbb{R}^{n}}Qe^{nu}dx<\\infty,$$ where $Q\\geq 0$, $\\kappa\\in (0,\\infty)$ and $n\\geq 3$. Using ODE techniques Martinazzi for $n=6$ and Huang-Ye for $n=4m+2$ proved the existence of solution to the above problem with $Q\\equiv const>0$ and for every $\\kappa\\in (0,\\infty)$. We extend these results in every dimension $n\\geq 5$, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which $Q$ is non-constant, and under some d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.01905","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":"1608.01905","created_at":"2026-05-18T00:42:32.616741+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.01905v1","created_at":"2026-05-18T00:42:32.616741+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.01905","created_at":"2026-05-18T00:42:32.616741+00:00"},{"alias_kind":"pith_short_12","alias_value":"IB27PIQJG5GF","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_16","alias_value":"IB27PIQJG5GFKFVH","created_at":"2026-05-18T12:30:22.444734+00:00"},{"alias_kind":"pith_short_8","alias_value":"IB27PIQJ","created_at":"2026-05-18T12:30:22.444734+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/IB27PIQJG5GFKFVH4VWXE6PLRA","json":"https://pith.science/pith/IB27PIQJG5GFKFVH4VWXE6PLRA.json","graph_json":"https://pith.science/api/pith-number/IB27PIQJG5GFKFVH4VWXE6PLRA/graph.json","events_json":"https://pith.science/api/pith-number/IB27PIQJG5GFKFVH4VWXE6PLRA/events.json","paper":"https://pith.science/paper/IB27PIQJ"},"agent_actions":{"view_html":"https://pith.science/pith/IB27PIQJG5GFKFVH4VWXE6PLRA","download_json":"https://pith.science/pith/IB27PIQJG5GFKFVH4VWXE6PLRA.json","view_paper":"https://pith.science/paper/IB27PIQJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.01905&json=true","fetch_graph":"https://pith.science/api/pith-number/IB27PIQJG5GFKFVH4VWXE6PLRA/graph.json","fetch_events":"https://pith.science/api/pith-number/IB27PIQJG5GFKFVH4VWXE6PLRA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IB27PIQJG5GFKFVH4VWXE6PLRA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IB27PIQJG5GFKFVH4VWXE6PLRA/action/storage_attestation","attest_author":"https://pith.science/pith/IB27PIQJG5GFKFVH4VWXE6PLRA/action/author_attestation","sign_citation":"https://pith.science/pith/IB27PIQJG5GFKFVH4VWXE6PLRA/action/citation_signature","submit_replication":"https://pith.science/pith/IB27PIQJG5GFKFVH4VWXE6PLRA/action/replication_record"}},"created_at":"2026-05-18T00:42:32.616741+00:00","updated_at":"2026-05-18T00:42:32.616741+00:00"}