{"paper":{"title":"Analysis of singularities of area-minimizing currents, Part III: branch points of planar frequency $\\neq$ 2, higher order asymptotics, and the local topology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Brian Krummel, Neshan Wickramasekera","submitted_at":"2026-07-15T00:43:59Z","abstract_excerpt":"This is the third part in a series of papers developing a new framework to study the local structure of $n$-dimensional area-minimizing rectifiable currents $T$ of codimension $\\geq 2$. Parts I and II introduced an intrinsic frequency function for $T$ -- planar frequency -- and used its monotonicity properties, among other things, to establish that ${\\mathcal H}^{n-2}$-a.e. branch point is a rapid-decay branch point where the planar frequency is at least $1 + \\alpha$. This paper analyses branch points of planar frequency $\\neq 2$. It establishes: (1) higher order asymptotics: at ${\\mathcal H}^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13356","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/2607.13356/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"}