{"paper":{"title":"Sobolev homeomorphisms and Brennan's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alexander Ukhlov, Vladimir Gol'dshtein","submitted_at":"2013-09-08T08:37:32Z","abstract_excerpt":"Let $\\Omega \\subset \\mathbb{R}^n$ be a domain that supports the $p$-Poincar\\'e inequality. Given a homeomorphism $\\varphi \\in L^1_p(\\Omega)$, for $p>n$ we show the domain $\\varphi(\\Omega)$ has finite geodesic diameter. This result has a direct application to Brennan's conjecture and quasiconformal homeomorphisms. {\\bf The Inverse Brennan's conjecture} states that for any simply connected plane domain $\\Omega' \\subset\\mathbb C$ with nonempty boundary and for any conformal homeomorphism $\\varphi$ from the unit disc $\\mathbb{D}$ onto $\\Omega'$ the complex derivative $\\varphi'$ is integrable in th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1940","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":""},"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"}