{"paper":{"title":"Approximation of solutions of the sinh-Gordon equation $\\Delta u -\\sinh(2u)=0$ by hyperbolic orthogonal ring patterns","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.MG","authors_text":"Ulrike B\\\"ucking","submitted_at":"2026-07-15T20:23:49Z","abstract_excerpt":"We consider hyperbolic orthogonal ring patterns as introduced in arXiv:2409.06573 and focus on their characterization by uniformizing variables at the centers of the rings. Given a smooth solution of the sinh-Gordon equation $\\Delta u -\\sinh(2u)=0$, we restrict to a compact subset of its domain and discretize it by square grid lattices with edge length $\\varepsilon$. Taking the values of $u$ as Dirichlet boundary conditions, we prove that the corresponding uniformizing variables $u^\\varepsilon$ of the hyperbolic ring patterns converge to $u$ in $C^\\infty$ with error of order $\\varepsilon^2$, g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14348","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.14348/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}