{"paper":{"title":"Hearing the shape of ancient noncollapsed flows in $\\mathbb{R}^{4}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Robert Haslhofer, Wenkui Du","submitted_at":"2021-07-09T13:46:45Z","abstract_excerpt":"We consider ancient noncollapsed mean curvature flows in $\\mathbb{R}^4$ whose tangent flow at $-\\infty$ is a bubble-sheet. We carry out a fine spectral analysis for the bubble-sheet function $u$ that measures the deviation of the renormalized flow from the round cylinder $\\mathbb{R}^2 \\times S^1(\\sqrt{2})$ and prove that for $\\tau\\to -\\infty$ we have the fine asymptotics $u(y,\\theta,\\tau)= (y^\\top Qy -2\\textrm{tr}(Q))/|\\tau| + o(|\\tau|^{-1})$, where $Q=Q(\\tau)$ is a symmetric $2\\times 2$-matrix whose eigenvalues are quantized to be either 0 or $-1/\\sqrt{8}$. This naturally breaks up the classi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2107.04443","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/2107.04443/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"}