{"paper":{"title":"A sharp relative comparison inequality for conformal fillings of Poincar\\'e--Einstein manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Nan Wu","submitted_at":"2026-07-15T11:59:46Z","abstract_excerpt":"Let $(X^{n+1},g_+)$ be a Poincar\\'e--Einstein manifold with conformal infinity $(M^n,[h])$ of positive Yamabe type. We prove the sharp relative comparison inequality $$\\frac{Y_1(X,M,[\\bar g])}{Y_1(\\mathbb{S}^{n+1}_+,\\mathbb{S}^n,[g_{\\mathbb{S}_+^{n+1}}])}\n  \\geq\n  \\left(\\frac{Y(M,[h])}{Y(\\mathbb{S}^n,[g_{\\mathbb{S}^n}])}\\right)^{\\frac{n}{n+1}} $$\n  for the type-I Escobar--Yamabe compactification, and establish the rigidity. This confirms a conjecture proposed by Sun-Yung A. Chang."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13742","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.13742/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}