{"paper":{"title":"On the size of the singular set of minimizing harmonic maps into the 2-sphere in dimension four and higher","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Armin Schikorra, Katarzyna Mazowiecka, Micha{\\l} Mi\\'skiewicz","submitted_at":"2019-02-08T16:04:10Z","abstract_excerpt":"We extend the results of our recent preprint [arXiv: 1811.00515] into higher dimensions $n \\geq 4$. For minimizing harmonic maps $u\\in W^{1,2}(\\Omega,\\mathbb{S}^2)$ from $n$-dimensional domains into the two dimensional sphere we prove:\n  (1) An extension of Almgren and Lieb's linear law, namely \\[\\mathcal{H}^{n-3}(\\textrm{sing} u) \\le C \\int_{\\partial \\Omega} |\\nabla_T u|^{n-1} \\,d\\mathcal{H}^{n-1};\\]\n  (2) An extension of Hardt and Lin's stability theorem, namely that the size of singular set is stable under small perturbations in $W^{1,n-1}$ norm of the boundary."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.03161","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"}