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Let $\\widetilde{M}$ be the universal covering of $M.$ Let $% \\Phi $ denote a $CR$ developing map {equation*} \\Phi :\\widetilde{M}\\rightarrow S^{2n+1} {equation*}% where $S^{2n+1}$ is the standard unit sphere in complex $n+1$-space $C^{n+1}$% . Suppose that the $CR$ Yamabe invariant of $M$ is positive. Then we show that $\\Phi $ is injective for $n\\geq 3$. In the case $n=2$, we also show that $\\Phi $ is injective under the condition: $s(M)<1$. It then follows that $M$ is uniformizable."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1301.1133","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-01-07T09:20:30Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"e15622d8cdc1e6804f42e5bbbc17253bd9c280ab81af12c0180fd6f266cf316b","abstract_canon_sha256":"30f446108901bf73b870c2f5cb33879bfa3c9b0543202e273652fb2af4344e54"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:06.765781Z","signature_b64":"bBtgZ42A14ZXhyUW3S5S3bjnDjjlu6qWLND1TDUaz/zRMW0ONCaGuFTa/1dIzxezmvV3PocSFhNK8LcB5DDSBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4213d4121537c646363e55ff6ac971cbdb044a2b5e63cd77f9be0445e7e9aa1b","last_reissued_at":"2026-05-18T03:37:06.764963Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:06.764963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniformization of spherical CR manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DG","authors_text":"Hung-Lin Chiu, Jih-Hsin Cheng, Paul Yang","submitted_at":"2013-01-07T09:20:30Z","abstract_excerpt":"Let $M$ be a closed (compact with no boundary) spherical $CR$ manifold of dimension $2n+1$. Let $\\widetilde{M}$ be the universal covering of $M.$ Let $% \\Phi $ denote a $CR$ developing map {equation*} \\Phi :\\widetilde{M}\\rightarrow S^{2n+1} {equation*}% where $S^{2n+1}$ is the standard unit sphere in complex $n+1$-space $C^{n+1}$% . Suppose that the $CR$ Yamabe invariant of $M$ is positive. Then we show that $\\Phi $ is injective for $n\\geq 3$. In the case $n=2$, we also show that $\\Phi $ is injective under the condition: $s(M)<1$. 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