{"paper":{"title":"On uniqueness and nonuniqueness of ancient ovals","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":"2021-05-28T13:45:29Z","abstract_excerpt":"In this paper, we prove that any nontrivial $\\mathrm{SO}(k )\\times \\mathrm{SO}(n+1-k)$-symmetric ancient compact noncollapsed solution of the mean curvature flow agrees up to scaling and rigid motion with the $\\mathrm{O}(k)\\times \\mathrm{O}(n+1-k)$-symmetric ancient ovals constructed by Hershkovits and the second author. This confirms a conjecture by Angenent-Daskalopoulos-Sesum. On the other hand, for every $k\\geq 2$ we also construct a $(k-1)$-parameter family of uniformly $(k+1)$-convex ancient ovals that are only $\\mathbb{Z}^{k}_{2}\\times \\mathrm{O}(n+1-k)$-symmetric. This gives counterexa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.13830","kind":"arxiv","version":2},"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/2105.13830/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"}