{"paper":{"title":"A sharp quantitative version of Alexandrov's theorem via the method of moving planes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.DG","authors_text":"Giulio Ciraolo, Luigi Vezzoni","submitted_at":"2015-01-30T17:05:25Z","abstract_excerpt":"We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let $S$ be a $C^2$ closed embedded hypersurface of $\\mathbb{R}^{n+1}$, $n\\geq1$, and denote by $osc(H)$ the oscillation of its mean curvature. We prove that there exists a positive $\\varepsilon$, depending on $n$ and upper bounds on the area and the $C^2$-regularity of $S$, such that if $osc(H) \\leq \\varepsilon$ then there exist two concentric balls $B_{r_i}$ and $B_{r_e}$ such that $S \\subset \\overline{B}_{r_e} \\setminus B_{r_i}$ and $r_e -r_i \\leq C \\, osc(H)$, with $C$ depending only on $n$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.07845","kind":"arxiv","version":3},"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"}