{"paper":{"title":"Contracting the boundary of a Riemannian 2-disc","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Alexander Nabutovsky, Regina Rotman, Yevgeny Liokumovich","submitted_at":"2012-05-24T14:57:17Z","abstract_excerpt":"Let $D$ be a Riemannian 2-disc of area $A$, diameter $d$ and length of the boundary $L$. We prove that it is possible to contract the boundary of $D$ through curves of length $\\leq L + 200d\\max\\{1,\\ln {\\sqrt{A}\\over d} \\}$. This answers a twenty-year old question of S. Frankel and M. Katz, a version of which was asked earlier by M.Gromov.\n  We also prove that a Riemannian $2$-sphere $M$ of diameter $d$ and area $A$ can be swept out by loops based at any prescribed point $p\\in M$ of length $\\leq 200 d\\max\\{1,\\ln{\\sqrt{A}\\over d} \\}$. This estimate is optimal up to a constant factor. In addition"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5474","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":""},"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"}