{"paper":{"title":"Euclidean Domains with Nearly Maximal Yamabe Quotient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Liam Mazurowski, Xuan Yao","submitted_at":"2025-01-21T18:25:31Z","abstract_excerpt":"Let $\\Omega$ be a smooth, bounded domain in $\\mathbb R^3$ with connected boundary. It follows from work of Escobar that the Yamabe quotient of $\\Omega$ is at most the Yamabe quotient of a ball, and equality holds if and only if $\\Omega$ is a ball. We show that if equality almost holds then the following things are true:\n  (i)$\\Omega$ is diffeomorphic to a ball;\n  (ii) There is a small number $\\epsilon > 0$ such that $B(x,r) \\subset \\Omega \\subset B(x,r(1+\\epsilon))$; (iii) After suitable scaling, $\\Omega$ is Gromov-Hausdorff close to the unit ball when considered as a metric space with its ind"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.12347","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/2501.12347/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"}