{"paper":{"title":"The $\\sigma_k$-Yamabe problem revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"If a closed manifold has positive Yamabe constant and positive σ₂-Yamabe constant, then the latter is achieved by a conformal metric.","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Guofang Wang, Wei Wei, Yuxin Ge","submitted_at":"2026-05-06T20:13:50Z","abstract_excerpt":"In this paper we revisit the $\\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\\sigma_k$-scalar curvature. We prove that on a closed manifold $\\left(M,\\left[g_0\\right]\\right)$ with positive Yamabe constant $Y_1\\left(M,\\left[g_0\\right]\\right)>0$, the $\\sigma_2$-Yamabe constant\n  $$ Y_2\\left(M,\\left[g_0\\right]\\right):=\\inf _{g \\in\\left[g_0\\right], R_g>0} \\frac{\\int_M \\sigma_2(g) d \\operatorname{vol}(g)}{\\operatorname{vol}(g)^{\\frac{n-4}{n}}} $$\n  is achieved by a conformal metric $g \\in\\left[g_0\\right]$, which in particular solves the $\\sigma_2$-Yamabe problem"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that on a closed manifold (M,[g₀]) with positive Yamabe constant Y₁(M,[g₀])>0, the σ₂-Yamabe constant Y₂(M,[g₀]) is achieved by a conformal metric g ∈ [g₀], which in particular solves the σ₂-Yamabe problem, assuming Y₂(M,[g₀])>0.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that Y₂(M,[g₀]) > 0 together with the restriction to metrics with R_g > 0 in the definition of the infimum; without R_g > 0 the conclusions can fail as shown in the paper.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"On closed manifolds with Y₁ > 0 and Y₂ > 0, the σ₂-Yamabe constant is achieved by a conformal metric with positive scalar curvature, and the two infima coincide.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If a closed manifold has positive Yamabe constant and positive σ₂-Yamabe constant, then the latter is achieved by a conformal metric.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7a923ca11e2be52eaab668d6eab80802903f2b8d57d79c37875c79749f6709f0"},"source":{"id":"2605.05414","kind":"arxiv","version":2},"verdict":{"id":"11595390-8fda-49a2-8a70-96879bbb43fb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-08T15:44:41.353155Z","strongest_claim":"We prove that on a closed manifold (M,[g₀]) with positive Yamabe constant Y₁(M,[g₀])>0, the σ₂-Yamabe constant Y₂(M,[g₀]) is achieved by a conformal metric g ∈ [g₀], which in particular solves the σ₂-Yamabe problem, assuming Y₂(M,[g₀])>0.","one_line_summary":"On closed manifolds with Y₁ > 0 and Y₂ > 0, the σ₂-Yamabe constant is achieved by a conformal metric with positive scalar curvature, and the two infima coincide.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that Y₂(M,[g₀]) > 0 together with the restriction to metrics with R_g > 0 in the definition of the infimum; without R_g > 0 the conclusions can fail as shown in the paper.","pith_extraction_headline":"If a closed manifold has positive Yamabe constant and positive σ₂-Yamabe constant, then the latter is achieved by a conformal metric."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.05414/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T21:01:19.350346Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T13:35:47.869737Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"7796eb7477c308107b22942cea92236b134bb9048b4b636836921fd596c63da4"},"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"}