{"paper":{"title":"Free boundary minimal surfaces: a nonlocal approach","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessandro Pigati, Francesca Da Lio","submitted_at":"2017-12-13T10:07:56Z","abstract_excerpt":"Given a $C^k$-smooth closed embedded manifold $\\mathcal N\\subset{\\mathbb R}^m$, with $k\\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\\partial S\\neq\\emptyset$, we consider $\\frac 12$-harmonic maps $u\\in H^{1/2}(\\partial S,\\mathcal N)$. These maps are critical points of the nonlocal energy \\begin{equation}E(f;g):=\\int_S\\big|\\nabla\\widetilde u\\big|^2\\,d\\text{vol}_g,\\end{equation} where $\\widetilde u$ is the harmonic extension of $u$ in $S$. We express the energy as a sum of the $\\frac 12$-energies at each boundary component of $\\partial S$ (suitably identified with the c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04683","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"}