{"paper":{"title":"Symmetry and Rigidity Results for the Mean Field Equation and Hawking Mass on ( \\mathbb{S}^2 )","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Solutions of the mean field equation on the sphere are symmetric for 1/3 ≤ α < 1/2, forcing rigidity of the Hawking mass for stable CMC spheres.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Amir Moradifam, Changfeng Gui","submitted_at":"2026-05-14T22:16:55Z","abstract_excerpt":"In this paper, we establish symmetry results for solutions of the mean field equation \\[ \\frac{\\alpha}{2} \\Delta u + e^u - 1 = 0 \\] on \\( \\mathbb{S}^2 \\) for $\\frac{1}{3}\\leq \\alpha < \\frac{1}{2}$.The proofs utilize the Sphere Covering Inequality and incorporate topological arguments on \\( \\mathbb{S}^2 \\). These results are further applied to demonstrate a rigidity property of the Hawking mass for stable constant mean curvature (CMC) spheres, addressing a question posed by Robert Bartnik in 2002. Our result unify and extend previous results on the rigidity of the Hawking mass for stable CMC sp"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Symmetry results for solutions of the mean field equation (α/2) Δu + e^u - 1 = 0 on S^2 for 1/3 ≤ α < 1/2, applied to demonstrate rigidity of the Hawking mass for stable CMC spheres, unifying and extending previous results for surfaces that are not nearly spherical.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Sphere Covering Inequality together with topological arguments on S^2 suffice to establish the claimed symmetry for the full range 1/3 ≤ α < 1/2 (abstract, paragraph on proofs).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Symmetry results for the mean field equation on S^2 for 1/3 ≤ α < 1/2 are established via the Sphere Covering Inequality and topology, then used to prove Hawking mass rigidity for stable CMC spheres.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Solutions of the mean field equation on the sphere are symmetric for 1/3 ≤ α < 1/2, forcing rigidity of the Hawking mass for stable CMC spheres.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"af0deb002930ee991bcd6ad0d3bbc1d7598b2d176bb1b70d560634e9a9423c71"},"source":{"id":"2605.15448","kind":"arxiv","version":1},"verdict":{"id":"a0226fda-bebc-4a60-af05-97c889ca2c91","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T14:38:56.733864Z","strongest_claim":"Symmetry results for solutions of the mean field equation (α/2) Δu + e^u - 1 = 0 on S^2 for 1/3 ≤ α < 1/2, applied to demonstrate rigidity of the Hawking mass for stable CMC spheres, unifying and extending previous results for surfaces that are not nearly spherical.","one_line_summary":"Symmetry results for the mean field equation on S^2 for 1/3 ≤ α < 1/2 are established via the Sphere Covering Inequality and topology, then used to prove Hawking mass rigidity for stable CMC spheres.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Sphere Covering Inequality together with topological arguments on S^2 suffice to establish the claimed symmetry for the full range 1/3 ≤ α < 1/2 (abstract, paragraph on proofs).","pith_extraction_headline":"Solutions of the mean field equation on the sphere are symmetric for 1/3 ≤ α < 1/2, forcing rigidity of the Hawking mass for stable CMC spheres."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15448/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"cited_work_retraction","ran_at":"2026-05-19T15:51:56.343631Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T15:50:05.282796Z","status":"completed","version":"0.1.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T15:01:17.622721Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T14:50:17.421586Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.112867Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.679764Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"8353264e1c59ea3209414559acef51041a16bc6dfbd0ea8cd17f2d7455edc8e3"},"references":{"count":24,"sample":[{"doi":"","year":1979,"title":"Meilleures constantes dans le th´ eor` eme d’inclusion de Sobolev et un th´ eor` eme de Fredholm non lin´ eaire pour la transformation conforme de la courbure scalaire.J","work_id":"a6bf34f3-7548-4643-99b1-3fcbc0d4cfcf","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"Mass and 3-metrics of non-negative scalar curvature","work_id":"7d260b1e-cffd-4458-b65b-370bbab08456","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1922,"title":"A singular sphere covering inequality: uniqueness and symmetry of solutions to singular Liouville- type equations.Math","work_id":"b1cb5ded-a667-4b98-9b26-c3496a2c35ba","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Localizing solutions of the Einstein constraint equations.Invent","work_id":"a04cc02e-3411-4ab5-9fc7-182490b9e9f5","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1988,"title":"Chang and Paul C","work_id":"fc2b7198-ea95-4793-b8f1-361f4ef780cc","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":24,"snapshot_sha256":"da7f923ae8417f13763b8b2bcb236571ca5ae1b229c3284dea0ea6f486a90a48","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"2ae5916e05b9310178427359020f83a7c38e119e6013a114802d465330e4999e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}