{"paper":{"title":"Minimal Graphs and Graphical Mean Curvature Flow in $M \\times \\mathbb R$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Ling Xiao, Matthew McGonagle","submitted_at":"2013-11-14T23:45:47Z","abstract_excerpt":"In this paper, we investigate the problem of finding minimal graphs in $M^n\\times\\mathbb R$ with general boundary conditions using a variational approach. We look at so called generalized solutions of the Dirichlet Problem that minimize a functional adapted from the area functional. We construct barriers to show that for certain conditions on our boundary data, $\\phi(x)$, the solutions obtain the boundary data $\\phi(x)$. Following Oliker-Ural'tseva we also consider solutions $u^{\\epsilon}$ of a perturbed mean curvature flow for $\\epsilon > 0$. We show that there are subsequences $\\epsilon_i$ w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.3699","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"}