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We prove that the Bartnik mass of the triple $(\\mathbb{S}^2,g,H)$ is bounded above by $\\sqrt{|\\mathbb{S}^2|_g/16\\pi}$ provided either (a) $\\lambda_1(g)=0$ and $H>0$; or (b) $\\lambda_1(g)>0$ and $H\\geq 0$. The eigenvalue condition, in particular, imposes no lower bound on $K_g$ (even under an area constraint) and thereby extends previous results which assume $K_g\\geq 0$."},"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":"2606.21816","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2026-06-20T00:41:15Z","cross_cats_sorted":[],"title_canon_sha256":"d6539ff977917daab777d54d3a8899c0053b3247a578868d3b8851e3fe0d4643","abstract_canon_sha256":"c159d4cb156281258ffb96b1585ec529b0d5654392687611cd45811fb4a6ee3b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T01:13:23.553991Z","signature_b64":"DQi1WuuOlhv+qLyOGonKKolisIMC6Tz7Urn9Yuu5+B354IhJbbkLV8DNlF6x+LQQKWgmR74Q6lTLD9pbEEPgDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a673b11d2fe18735b5f6640c450165e74a00a77ecbf4662e4553b24f5e859e80","last_reissued_at":"2026-06-23T01:13:23.553540Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T01:13:23.553540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bartnik Mass of CMC surfaces under a Spectral non-negativity condition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Albert Chau, Luke Kuo Han","submitted_at":"2026-06-20T00:41:15Z","abstract_excerpt":"Let \\(g\\) be a smooth Riemannian metric and $H$ a function on $\\mathbb{S}^2$ and let $\\lambda_1(g)$ be the first eigenvalue of the operator $(-\\Delta_g+K_g)$. We prove that the Bartnik mass of the triple $(\\mathbb{S}^2,g,H)$ is bounded above by $\\sqrt{|\\mathbb{S}^2|_g/16\\pi}$ provided either (a) $\\lambda_1(g)=0$ and $H>0$; or (b) $\\lambda_1(g)>0$ and $H\\geq 0$. 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