{"paper":{"title":"The boundary behavior of domains with complete translating, minimal and CMC graphs in $N^2\\times \\mathbb{R}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hengyu Zhou","submitted_at":"2017-09-10T14:16:21Z","abstract_excerpt":"In this note we discuss graphs over a domain $\\Omega\\subset N^2$ in the product manifold $N^2\\times \\mathbb{R}$. Here $N^2$ is a complete Riemannian surface and $\\Omega$ has peice-wise smooth boundary. Let $\\gamma \\subset\\partial\\Omega$ be a smooth connected arc and $\\Sigma$ be a complete graph in $N^2\\times \\mathbb{R}$ over $\\Omega$. We show that if $\\Sigma$ is a minimal or translating graph, then $\\gamma$ is a geodesic in $N^2$. Moreover if $\\Sigma$ is a CMC graph, then $\\gamma$ has constant principle curvature in $N^2$. This explains the infinity value boundary condition upon domains having"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.03104","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"}