{"paper":{"title":"Uniformisation of a once-punctured annulus","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Tanran Zhang","submitted_at":"2014-01-30T10:31:49Z","abstract_excerpt":"The universal cover or the covering group of a hyperbolic Riemann surface $X$ is important but hard to express explicitly. It can be, however, detected by the uniformisation and a suitable description of $X$. Beardon proposed five different ways to describe twice-punctured disks using fundamental domain, hyperbolic length, collar and extremal length in 2012. We parameterize a once-punctured annulus $A$ in terms of five parameter pairs and give explicit formulas about the hyperbolic structure and the complex structure of $A$. Several degenerating cases are also treated."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.7786","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"}