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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"},"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":"2605.05414","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2026-05-06T20:13:50Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"04a2362296a64b2527e144a0be13529c73555ccd58cf2d28f495589717d00883","abstract_canon_sha256":"ee114037d0f7007381c99fc67cbed061199fb9ddd7251e3f9bf73654e1d554c0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:05:46.046817Z","signature_b64":"jxyiLX0K5bLyplPSM+ABgV8xY5mrKs44D13hjMdmJMVImZL+Db6LOtf0EiaxSJ2l9VpCFrkRpwIUQTaSMRomCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8da706a240dd23e819cc29b48e3bab75523f04fa40db4865a29afbf8e6cc0c3e","last_reissued_at":"2026-05-20T00:05:46.046040Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:05:46.046040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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; 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