{"paper":{"title":"Uniqueness of immersed spheres in three-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Jose A. Galvez, Pablo Mira","submitted_at":"2016-03-23T12:15:31Z","abstract_excerpt":"Let $\\mathcal{A}$ be a class of immersed surfaces in a three-manifold $M$, and assume that $\\mathcal{A}$ is modeled by an elliptic PDE over each tangent plane. In this paper we solve the so-called Hopf uniqueness problem for the class $\\mathcal{A}$ under the only mild assumption of the existence of a transitive family of candidate surfaces $\\mathcal{S}\\subset \\mathcal{A}$. Specifically, we prove that any compact immersed surface of genus zero in the class $\\mathcal{A}$ is a candidate sphere. This theorem unifies and extends many previous uniqueness results of different contexts. As an applicat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.07153","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"}