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Define a functional $$J_{\\alpha,\\beta}(u)=\\frac{1}{2}\\int_\\Sigma(|\\nabla_gu|^2-\\alpha u^2)dv_g-\\beta\\log\\int_\\Sigma he^udv_g$$ on a function space $\\mathcal{H}=\\left\\{u\\in W^{1,2}(\\Sigma): \\int_\\Sigma udv_g=0\\right\\}$. If $\\alpha<\\lambda_1(\\Sigma)$ and $J_{\\alpha,8\\pi}$ has no minimizer on $\\mathcal{H}$, then we calculate the infimum of $J_{\\alpha,8\\pi}$ on $\\mathcal{H}$ by using the met"},"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":"1706.08207","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2017-06-26T02:20:21Z","cross_cats_sorted":[],"title_canon_sha256":"3485ab10317a67505ef02ee68b7fdd8837e4b7d160e5af97677a6845c5e3531f","abstract_canon_sha256":"d713d35c613fbfc9cd675ce4bb7a15423e7e0043fe8cfcdf86528f7c045a784d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:30.778306Z","signature_b64":"YQEnSt1OMIMoqV6/aIje/FRGqE7Ao2EKkHRUT88cRhKUdEV5MVgQP1Qng/95ZZc9Ggj1XucGoSo4cA8IVu5XAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f39c097fe87fa857a473504a7ee28c1fd281a92c76b23ef964a9b64afd0f68a","last_reissued_at":"2026-05-18T00:32:30.777526Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:30.777526Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence of solutions to a class of Kazdan-Warner equations on compact Riemannian surface","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaobao Zhu, Yunyan Yang","submitted_at":"2017-06-26T02:20:21Z","abstract_excerpt":"Let $(\\Sigma,g)$ be a compact Riemannian surface without boundary and $\\lambda_1(\\Sigma)$ be the first eigenvalue of the Laplace-Beltrami operator $\\Delta_g$. Let $h$ be a positive smooth function on $\\Sigma$. Define a functional $$J_{\\alpha,\\beta}(u)=\\frac{1}{2}\\int_\\Sigma(|\\nabla_gu|^2-\\alpha u^2)dv_g-\\beta\\log\\int_\\Sigma he^udv_g$$ on a function space $\\mathcal{H}=\\left\\{u\\in W^{1,2}(\\Sigma): \\int_\\Sigma udv_g=0\\right\\}$. If $\\alpha<\\lambda_1(\\Sigma)$ and $J_{\\alpha,8\\pi}$ has no minimizer on $\\mathcal{H}$, then we calculate the infimum of $J_{\\alpha,8\\pi}$ on $\\mathcal{H}$ by using the met"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.08207","kind":"arxiv","version":2},"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":"1706.08207","created_at":"2026-05-18T00:32:30.777644+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.08207v2","created_at":"2026-05-18T00:32:30.777644+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.08207","created_at":"2026-05-18T00:32:30.777644+00:00"},{"alias_kind":"pith_short_12","alias_value":"P444BF76Q75I","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_16","alias_value":"P444BF76Q75IK6SH","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_8","alias_value":"P444BF76","created_at":"2026-05-18T12:31:37.085036+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/P444BF76Q75IK6SHGUCKP3RIYH","json":"https://pith.science/pith/P444BF76Q75IK6SHGUCKP3RIYH.json","graph_json":"https://pith.science/api/pith-number/P444BF76Q75IK6SHGUCKP3RIYH/graph.json","events_json":"https://pith.science/api/pith-number/P444BF76Q75IK6SHGUCKP3RIYH/events.json","paper":"https://pith.science/paper/P444BF76"},"agent_actions":{"view_html":"https://pith.science/pith/P444BF76Q75IK6SHGUCKP3RIYH","download_json":"https://pith.science/pith/P444BF76Q75IK6SHGUCKP3RIYH.json","view_paper":"https://pith.science/paper/P444BF76","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.08207&json=true","fetch_graph":"https://pith.science/api/pith-number/P444BF76Q75IK6SHGUCKP3RIYH/graph.json","fetch_events":"https://pith.science/api/pith-number/P444BF76Q75IK6SHGUCKP3RIYH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P444BF76Q75IK6SHGUCKP3RIYH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P444BF76Q75IK6SHGUCKP3RIYH/action/storage_attestation","attest_author":"https://pith.science/pith/P444BF76Q75IK6SHGUCKP3RIYH/action/author_attestation","sign_citation":"https://pith.science/pith/P444BF76Q75IK6SHGUCKP3RIYH/action/citation_signature","submit_replication":"https://pith.science/pith/P444BF76Q75IK6SHGUCKP3RIYH/action/replication_record"}},"created_at":"2026-05-18T00:32:30.777644+00:00","updated_at":"2026-05-18T00:32:30.777644+00:00"}