{"paper":{"title":"Singular Yamabe problem for scalar flat metrics on the sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram Karakhanyan","submitted_at":"2017-11-05T22:16:25Z","abstract_excerpt":"Let $\\Omega$ be a domain on the unit $n$-sphere $ \\mathbb S^n$ and $\\mathring{g}$ the standard metric of $\\mathbb S^n$, $n\\ge 3$. We show that there exists a conformal metric $g$ with vanishing scalar curvature $R(g)=0$ such that $(\\Omega, g)$ is complete if and only if the Bessel capacity $\\mathcal C_{\\alpha, q}(\\mathbb S^n\\setminus \\Omega)=0$, where $\\alpha=1+\\frac2n$ and $q=\\frac n2$. Our analysis utilizes some well known properties of capacity and Wolff potentials, as well as a version of the Hopf-Rinow theorem for the divergent curves."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.01669","kind":"arxiv","version":2},"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"}