{"paper":{"title":"Liouville's theorem in calibrated geometries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.SG"],"primary_cat":"math.DG","authors_text":"Pekka Pankka, Toni Ikonen","submitted_at":"2024-10-03T17:44:35Z","abstract_excerpt":"We consider the following extension of the classical Liouville theorem: A calibration $\\omega \\in \\Lambda^n \\mathbb{R}^m$, where $3 \\le n \\le m$, has the Liouville property if a Sobolev mapping $F\\colon \\Omega \\to \\mathbb{R}^m$, where $\\Omega \\subset \\mathbb{R}^n$ is a domain, in $W^{1,n}_{loc}( \\Omega, \\mathbb{R}^m )$ satisfying $\\|DF\\|^n = \\star F^{*}\\omega$ almost everywhere is a restriction of a M\\\"obius transformation $\\mathbb{S}^m \\to \\mathbb{S}^m$.\n  We show that, for $m\\ge 5$, every calibration in $\\Lambda^{m-2} \\mathbb{R}^m$ has the Liouville property and, in low dimensions, a calibra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.02722","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/2410.02722/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"}