{"paper":{"title":"On the Nature of Stationary Integral Varifolds near Multiplicity 2 Planes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Neshan Wickramasekera, Paul Minter, Spencer Becker-Kahn","submitted_at":"2025-07-17T14:13:29Z","abstract_excerpt":"We study stationary integral $n$-varifolds $V$ in the unit ball $B_1(0)\\subset\\mathbb{R}^{n+k}$. Allard's regularity theorem establishes the existence of $\\epsilon = \\epsilon(n,k)\\in (0,1)$ for which if $V$ is $\\epsilon$-close (as varifolds) to the plane $P_0 = \\{0\\}^k\\times\\mathbb{R}^n$ with multiplicity 1 then, in $B_{1/2}(0)$, $V$ is represented by a single $C^{1,\\alpha}$ minimal graph. However, when instead $P_0$ occurs with multiplicity $Q\\in \\{2,3,\\dotsc\\}$, simple examples show that this conclusion, now as a multi-valued graph, may fail, even if $V$ corresponds to an area-minimising rec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.13148","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/2507.13148/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"}