{"paper":{"title":"A proof of the Muir-Suffridge conjecture for convex maps of the unit ball in $\\mathbb C^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CV","authors_text":"Filippo Bracci, Herv\\'e Gaussier","submitted_at":"2017-01-20T15:56:05Z","abstract_excerpt":"We prove (and improve) the Muir-Suffridge conjecture for holomorphic convex maps. Namely, let $F:\\mathbb B^n\\to \\mathbb C^n$ be a univalent map from the unit ball whose image $D$ is convex. Let $\\mathcal S\\subset \\partial \\mathbb B^n$ be the set of points $\\xi$ such that $\\lim_{z\\to \\xi}\\|F(z)\\|=\\infty$. Then we prove that $\\mathcal S$ is either empty, or contains one or two points and $F$ extends as a homeomorphism $\\tilde{F}:\\overline{\\mathbb B^n}\\setminus \\mathcal S\\to \\overline{D}$. Moreover, $\\mathcal S=\\emptyset$ if $D$ is bounded, $\\mathcal S$ has one point if $D$ has one connected comp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05836","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"}