{"paper":{"title":"Combinatorial Harmonic Maps and Convergence to Conformal Maps, I: A Harmonic Conjugate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.GT","authors_text":"Sa'ar Hersonsky","submitted_at":"2012-08-13T20:23:53Z","abstract_excerpt":"In this paper, we provide new discrete uniformization theorems for bounded, $m$-connected planar domains. To this end, we consider a planar, bounded, $m$-connected domain $\\Omega$ and let $\\bord\\Omega$ be its boundary. Let $\\mathcal{T}$ denote a triangulation of $\\Omega\\cup\\bord\\Omega$. We construct a \\emph{new} decomposition of $\\Omega\\cup\\bord\\Omega$ into a finite union of quadrilaterals with disjoint interiors. The construction is based on utilizing a {\\it pair} of harmonic functions on ${\\mathcal T}^{(0)}$ and properties of their level curves. In the sequel \\cite{Her3} it will be proved th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2703","kind":"arxiv","version":5},"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"}