{"paper":{"title":"Density of a minimal submanifold and total curvature of its boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Jaigyoung Choe, Robert Gulliver","submitted_at":"2017-09-07T05:44:36Z","abstract_excerpt":"Given a piecewise smooth submanifold $\\Gamma^{n-1} \\subset \\R^m$ and $p \\in \\R^m$, we define the {\\em vision angle} $\\Pi_p(\\Gamma)$ to be the $(n-1)$-dimensional volume of the radial projection of $\\Gamma$ to the unit sphere centered at $p$. If $p$ is a point on a stationary $n$-rectifiable set $\\Sigma \\subset \\R^m$ with boundary $\\Gamma$, then we show the density of $\\Sigma$ at $p$ is $\\leq$ the density at its vertex $p$ of the cone over $\\Gamma$. It follows that if $\\Pi_p(\\Gamma)$ is less than twice the volume of $S^{n-1}$, for all $p \\in \\Gamma$, then $\\Sigma$ is an embedded submanifold. As"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02078","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"}